Diophantine approximation on planar curves and the distribution of rational points

Victor Beresnevich ^{∗}

MINSK

Detta Dickinson

MAYNOOTH

Sanju Velani^{†}

YORK

With an Appendix

Sums of two squares near perfect squares

R. C. Vaughan ^{‡}

PENN STATE

IN MEMORY OF PRITISH LIMANI (1983–2003)

Abstract

Let C be a non–degenerate planar curve and for a real, positive decreasing function ψ let C(ψ) denote the set of simultaneously ψ–approximable points lying on C. We show that C is of Khintchine type for divergence; i.e. if a certain sum diverges then the one-dimensional Lebesgue measure on C of C(ψ) is full. We also obtain the Hausdorff measure analogue of the divergent Khintchine type result. In the case that C is a rational quadric the convergence counterparts of the divergent results are also obtained. Furthermore, for functions ψ with lower order in a critical range we determine a general, exact formula for the Hausdorff dimension of C(ψ). These results constitute the first precise and general results in the theory of simultaneous Diophantine approximation on manifolds.

2000 Mathematics Subject Classification: Primary 11J83; Secondary 11J13, 11K60

Keywords and phrases: Diophantine approximation, Khintchine type theorems, Hausdorff measure and dimension, Distribution of rational points, Ubiquitous systems

∗This work has been partially supported by INTAS Project 00-429 and by EPSRC grant GR/R90727/01 †Royal Society University Research Fellow ‡Research supported by NSA grant MDA904-03-1-0082.

Contents

1. Introduction 3 1.1 Background and the general problems . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 The Khintchine type theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1. The Khintchine theory for rational quadrics . . . . . . . . . . . . . . . 6 1.3 The Hausdorff measure/dimension theory . . . . . . . . . . . . . . . . . . . . . 7 1.4 Rational points close to a curve . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2. Proof of the rational quadric statements 12 2.1 Proof of Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Hausdorff measure and dimension . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Proof of Theorem 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3. Ubiquitous systems 15 3.1 Ubiquitous systems in ℝ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2 Ubiquitous systems close to a curve in ℝ^{n} . . . . . . . . . . . . . . . . . . . . . 17

4. Proof of Theorem 6 19 4.1 The ubiquity version of Theorem 6 . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.2 An auxiluary lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.3 Proof of Theorem 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5. Proof of Theorem 4 23

6. Proof of Theorem 1 24

7. Proof of Theorem 3 27

8. Various Generalizations 28 8.1 Theorem 3 for general Hausdorff measures . . . . . . . . . . . . . . . . . . . . 28 8.2 The Multiplicative Problems/Theory . . . . . . . . . . . . . . . . . . . . . . . 29

APPENDIX I: Proof of Ubiquity Lemmas 30

APPENDIX II: Sums of Two Squares Near Perfect Squares 47

1. Introduction

In n–dimensional Euclidean space there are two main types of Diophantine approximation which can be considered, namely simultaneous and dual. Briefly, the simultaneous case involves approximating points y = (y_{1}, …, y_{n}) in ℝ^{n} by rational points {p/q : (p, q) ∈ ℤ^{n} × ℤ}. On the other hand, the dual case involves approximating points y by rational hyperplanes {q.x = p : (p, q) ∈ ℤ × ℤ^{n}} where x.y = x_{1}y_{1} + ⋯ + x_{n}y_{n} is the standard scalar product of two vectors x, y ∈ ℝ^{n}. In both cases the ‘rate’ of approximation is governed by some given approximating function. In this paper we consider the general problem of simultaneous Diophantine approximation on manifolds. Thus, the points in ℝ^{n} of interest are restricted to some manifold M embedded in ℝ^{n}. Over the past ten years or so, major advances have been made towards developing a complete ‘metric’ theory for the dual form of approximation. However, no such theory exists for the simultaneous case. To some extent this work is an attempt to address this in balance.

1.1 Background and the general problems

Simultaneous approximation in ℝ^{n}. In order to set the scene we recall two fundamental results in the theory of simultaneous Diophantine approximation in n–dimensional Euclidean space. Throughout ψ : ℝ^{+} → ℝ^{+} will denote a real, positive decreasing function and will be referred to as an approximating function. Given an approximating function ψ, a point y = (y_{1}, …, y_{n}) ∈ ℝ^{n} is called simultaneously ψ–approximable if there are infinitely many q ∈ ℕ such that

max_{1⩽i⩽n} ∥qy_{i}∥ < ψ(q)

where ∥x∥ = min{|x−m| : m ∈ ℤ}. In the case ψ is ψ_{v} : h → h^{−v} with v > 0 the point y is said to be simultaneously v–approximable. The set of simultaneously ψ–approximable points will be denoted by S_{n}(ψ) and similarly S_{n}(v) will denote the set of simultaneously v–approximable points in ℝ^{n}. Note that in view of Dirichlet’s theorem (n-dimensional simultaneous version), S_{n}(v) = ℝ^{n} for any v ≤ 1/n.

The following fundamental result provides a beautiful and simple criteria for the ‘size’ of the set S_{n}(ψ) expressed in terms of n–dimensional Lebesgue measure | |_{ℝ^{n}}.

Khintchine’s Theorem (1924). Let ψ be an approximating function. Then

|S_{n}(ψ)|_{ℝ^{n}} = \{ \begin{matrix}ZERO & if ∑ ψ(h)^{n} < ∞ \\ FULL & if ∑ ψ(h)^{n} = ∞ \end{matrix} .

Here ‘full’ simply means that the complement of the set under consideration is of zero measure. Thus the n–dimensional Lebesgue measure of the set of simultaneously ψ–approximable points in ℝ^{n} satisfies a ‘zero-full’ law. The divergence part of the above statement constitutes the main substance of the theorem. The convergence part is a simple consequence of the Borel-Cantelli lemma from probability theory. Note that |S_{n}(v)|_{ℝ^{n}} = 0 for v > 1/n and so ℝ^{n} is extremal – see below.

The next fundamental result is a Hausdorff measure version of the above theorem and shows that the s–dimensional Hausdorff measure H^{s}(S_{n}(ψ)) of the set S_{n}(ψ) satisfies an elegant ‘zero-infinity’ law.

Jarník’s Theorem (1931). Let s ∈ (0, n) and ψ be an approximating function. Then

H^{s} (S_{n}(ψ)) = \{ \begin{matrix}0 & if ∑ h^{n−s} ψ(h)^{s} < ∞ \\ ∞ & if ∑ h^{n−s} ψ(h)^{s} = ∞ \end{matrix} .

Furthermore

dim S_{n}(ψ) = inf{s : ∑ h^{n−s} ψ(h)^{s} < ∞} .

The dimension part of the statement follows directly from the definition of Hausdorff dimension – see §2.2. In Jarník’s original statement the additional hypotheses that rψ(r)^{n} → 0 as r → ∞, rψ(r)^{n} is decreasing and that r^{1+n−s}ψ(r)^{s} is decreasing were assumed. However, these are not necessary – see [6, §1.1 and §12.1]. Also, Jarník obtained his theorem for general Hausdorff measures H^{h} where h is a dimension function – see §8.1 and [6, §1.1 and §12.1]. However, for the sake of clarity and ease of discussion we have specialized to s-dimensional Hausdorff measure. Note that the above theorem implies that for v > 1/n

H^{d} (S_{n}(v)) = ∞  where  d := dim S_{n}(v) = \frac{1 + n}{v + 1} .

The two fundamental theorems stated above provide a complete measure theoretic description of S_{n}(ψ). For a more detailed discussion and various generalizations of these theorems see [6].

Simultaneous approximation restricted to manifolds. Let M be a manifold of dimension m embedded in ℝ^{n}. Given an approximating function ψ consider the set

M ∩ S_{n}(ψ)

consisting of points y on M which are simultaneously ψ–approximable. Two natural problems now arise.

Problem 1. To develop a Khintchine type theory for M ∩ S_{n}(ψ).

Problem 2. To develop a Hausdorff measure/dimension theory for M ∩ S_{n}(ψ).

In short, the aim is to establish analogues of the two fundamental theorems described above and thereby provide a complete measure theoretic description of the sets M∩S_{n}(ψ). The fact that the points y of interest are of dependent variables, reflecting the fact that y ∈ M introduces major difficulties in attempting to describe the measure theoretic structure of M∩S_{n}(ψ). This is true even in the specific case that M is a planar curve. More to the point, even for seemingly simple curves such as the unit circle or the parabola the problem is fraught with difficulties.

Non-degenerate manifolds. In order to make any reasonable progress with the above problems it is not unreasonable to assume that the manifolds M under consideration are nondegenerate [23]. Essentially, these are smooth sub-manifolds of ℝ^{n} which are sufficiently curved so as to deviate from any hyperplane. Formally, a manifold M of dimension m embedded in ℝ^{n} is said to be non-degenerate if it arises from a non–degenerate map f : U → ℝ^{n} where U is an open subset of ℝ^{m} and M := f(U). The map f : U → ℝ^{n} : u ↦ f(u) = (f_{1}(u), …, f_{n}(u)) is said to be non–degenerate at u ∈ U if there exists some l ∈ ℕ such that f is l times continuously differentiable on some sufficiently small ball centred at u and the partial derivatives of f at u of orders up to l span ℝ^{n}. The map f is non–degenerate if it is non–degenerate at almost every (in terms of m–dimensional Lebesgue measure) point in U; in turn the manifold M = f(U) is also said to be non–degenerate. Any real, connected analytic manifold not contained in any hyperplane of ℝ^{n} is non–degenerate.

Note that in the case the manifold M is a planar curve C, a point on C is non-degenerate if the curvature at that point is non-zero. Thus, C is a non-degenerate planar curve if the set of points on C at which the curvature vanishes is a set of one–dimensional Lebesgue measure zero. Moreover, it is not difficult to show that the set of points on a planar curve at which the curvature vanishes but the curve is non-degenerate is at most countable. In view of this, the curvature completely describes the non-degeneracy of planar curves. Clearly, a straight line is degenerate everywhere.

1.2 The Khintchine type theory

The aim is to obtain an analogue of Khintchine’s theorem for the set M ∩ S_{n}(ψ) of simultaneously ψ–approximable points lying on M. First of all notice that if the dimension m of the manifold M is strictly less than n then |M ∩ S_{n}(ψ)|_{ℝ^{n}} = 0 irrespective of the approximating function ψ. Thus, when referring to the Lebesgue measure of the set M ∩ S_{n}(ψ) it is always with reference to the induced Lebesgue measure on M. More generally, given a subset S of M we shall write |S|_{M} for the measure of S with respect to the induced Lebesgue measure on M. Notice that for v ≤ 1/n, we have that |M ∩ S_{n}(v)|_{M} = |M|_{M} := FULL as it should be since S_{n}(v) = ℝ^{n}.

To develop the Khintchine theory it is natural to consider the convergence and divergence cases separately and the following terminology is most useful.

Definition 1 Let M ⊂ ℝ^{n} be a manifold. Then we say that

1. M is of Khintchine type for convergence if |M ∩ S_{n}(ψ)|_{M} = ZERO for any approximating function ψ with ∑_{h=1}^{∞} ψ(h)^{n} < ∞.

2. M is of Khintchine type for divergence if |M ∩ S_{n}(ψ)|_{M} = FULL for any approximating function ψ with ∑_{h=1}^{∞} ψ(h)^{n} = ∞.

The set of manifolds which are of Khintchine type for convergence will be denoted by K_{<∞}. Similarly, the set of manifolds which are of Khintchine type for divergence will be denoted by K_{=∞}. Also, we define K := K_{<∞} ∩ K_{=∞}. By definition, if M ∈ K then an analogue of Khintchine’s theorem exists for M ∩ S_{n}(ψ) and M is simply said to be of Khintchine type. Thus Problem 1 mentioned above, is equivalent to describing the set of Khintchine type manifolds. Ideally, one would like to prove that any non-degenerate manifold is of Khintchine type. Similar terminology exists for the dual form of approximation in which ‘Khintchine type’ is replaced by ‘Groshev type’; for further details see [11, pp. 29–30].

A weaker notion than ‘Khintchine type for convergence’ is that of extremality. A manifold M is said to be extremal if |M ∩ S_{n}(v)|_{M} = 0 for any v > 1/n. The set of extremal manifolds of ℝ^{n} will be denoted by E and it is readily verified that K_{<∞} ⊂ E. In 1932, Mahler made the conjecture that for any n ∈ ℕ the Veronese curve V_{n} = {(x, x^{2}, …, x^{n}) : x ∈ ℝ} is extremal. The conjecture was eventually settled in 1964 by Sprindzuk [28] – the special cases n = 2 and 3 had been done earlier. Essentially, it is this conjecture and its investigations which gave rise to the now flourishing area of ‘Diophantine approximation on manifolds’ within metric number theory. Up to 1998, manifolds satisfying a variety of analytic, arithmetic and geometric constraints had been shown to be extremal. For example, Schmidt in 1964 proved that any C^{3} planar curve with non-zero curvature almost everywhere is extremal. However, Sprindzuk in the 1980’s, had conjectured that any analytic manifold satisfying a necessary non–degeneracy condition is extremal. In 1998, Kleinbock and Margulis [23] showed that any non-degenerate manifold is extremal and thereby settled the conjecture of Sprindzuk.

Regarding the ‘Khintchine theory’ very little is known. The situation for the dual form of approximation is very different. For the dual case, it has recently been shown that any non–degenerate manifold is of Groshev type – the analogue of Khintchine type in the dual case (see [5], [12] and [6, §12.7]). For the simultaneous case, the current state of the Khintchine theory is somewhat ad-hoc. Either a specific manifold or a special class of manifolds satisfying various constraints is studied. For example it has been shown that (i) manifolds which are a topological product of at least four non–degenerate planar curves are in K [8]; (ii) the parabola V_{2} is in K_{<∞} [9]; (iii) the so called 2–convex manifolds of dimension m ≥ 2 are in K_{<∞} [17] and (iv) straight lines through the origin satisfying a natural Diophantine condition are in K_{<∞} [24]. Thus, even in the simplest geometric and arithmetic situation in which the manifold is a genuine curve in ℝ^{2} the only known result to date is that of the parabola V_{2}. To our knowledge, no curve has ever been shown to be in K_{=∞}.

In this paper we address the fundamental problems of §1.1 in the case that the manifold M is a planar curve (the specific case that M is a non-degenerate, rational quadric will be shown in full). Regarding Problem 1, our main result is the following. As usual, C^{(n)}(I) will denote the set of n–times continuously differentiable functions defined on some interval I of ℝ.

Theorem 1 Let ψ be an approximating function with ∑_{h=1}^{∞} ψ(h)^{2} = ∞. Let f ∈ C^{(3)}(I_{0}), where I_{0} is an interval, and f′′(x) ≠ 0 for almost all x ∈ I_{0}. Then for almost all x ∈ I_{0} the point (x, f(x)) is simultaneously ψ–approximable.

Corollary 1 Any C^{(3)} non–degenerate planar curve is of Khintchine type for divergence.

To complete the ‘Khintchine theory’ for C^{(3)} non–degenerate planar curves we need to show that any such curve is of Khintchine type for convergence. We are currently able to prove this in the special case that the planar curve is a non-degenerate, rational quadric. However, the truth of Conjecture 1 in §1.4 regarding the distribution of rational points ‘near’ planar curves would yield the complete convergence theory.

1.2.1. The Khintchine theory for rational quadrics

As above, let V_{2} := {(x_{1}, x_{2}) ∈ ℝ^{2} : x_{2} = x_{1}^{2}} denote the standard parabola and let C_{1} := {(x_{1}, x_{2}) ∈ ℝ^{2} : x_{1}^{2} + x_{2}^{2} = 1} and C_{1}^{∗} := {(x_{1}, x_{2}) ∈ ℝ^{2} : x_{1}^{2} − x_{2}^{2} = 1} denote the unit circle and standard hyperbola respectively. Next, let Q denote a non-degenerate, rational quadric in the plane. By this we mean that Q is the image of either the circle C_{1}, the hyperbola C_{1}^{∗} or the parabola V_{2} under a rational affine transformation of the plane. Furthermore, for an approximating function ψ let

Q(ψ) := Q ∩ S_{2}(ψ)

In view of Corollary 1 we have that Q is in K_{=∞}. The following result shows that any nondegenerate, rational quadric is in fact in K and provides a complete criteria for the size of Q(ψ) expressed in terms of Lebesgue measure. Clearly, it contains the only previously known result that the parabola is in K_{<∞}.

Theorem 2 Let ψ be an approximating function. Then

|Q(ψ)|_{Q} = \{ \begin{matrix}ZERO & if ∑ ψ(h)^{2} < ∞ \\ FULL & if ∑ ψ(h)^{2} = ∞ \end{matrix} .

1.3 The Hausdorff measure/dimension theory

The aim is to obtain an analogue of Jarník’s theorem for the set M ∩ S_{n}(ψ) of simultaneously ψ–approximable points lying on M. In the dual case, the analogue of the divergent part of Jarník’s theorem has recently been established for any non-degenerate manifold [6, §12.7]. Prior to this, a general lower bound for the Hausdorff dimension of the dual set of v–approximable points lying on any extremal manifold had been obtained [13]. Also in the dual case, exact formulae for the dimension of the dual v-approximating sets are known for the case of the Veronese curve [2, 10] and for any planar curve with curvature non-zero except for a set of dimension zero [1].

As with the Khintchine theory, very little is currently known regarding the Hausdorff measure/dimension theory for the simultaneous case. Contrary to the dual case, dim M∩S_{n}(v) behaves in a rather complicated way and appears to depend on the arithmetic properties of M. For example, let C_{R} = {x^{2} +y^{2} = R^{2}} be the circle of radius R centered at the origin. It is easy to verify that C\sqrt{3} contains no rational points (s/q, t/q). On the other hand, any Pythagorean triple (s, t, q) gives rise to a rational point on the unit circle C_{1} and so there are plenty of rational points on C_{1}. For v > 1, these facts regarding the distribution of rational points on the circle under consideration lead to dim C\sqrt{3} ∩ S_{2}(v) = 0 whereas dim C_{1} ∩ S_{2}(v) = 1/(1 + v) [6, 14]. The point is that for v > 1, the rational points of interest must lie on the associated circle. Further evidence for the complicated behavior of the dimension can be found in [26]. Recently, dim M ∩ S_{n}(v) has been calculated for large values of v when the manifold M is parameterized by polynomials with integer coefficients [15] and for v > 1 when the manifold is a non-degenerate, rational quadric in ℝ^{n} [18]. Also, as a consequence of Wiles’ theorem [30], dim M ∩ S_{2}(v) = 0 for the curve x^{k} + y^{k} = 1 with k > 2 and v > k − 1 [11, p. 94].

The above examples illustrate that in the simultaneous case there is no hope of establishing a single, general formula for dim M ∩ S_{n}(v). Recall, that for v = 1/n we have that dim M ∩ S_{n}(v) = dim M := m for any manifold embedded in ℝ^{n} since S_{n}(v) = ℝ^{n} by Dirichlet’s theorem. Now notice that in the various examples considered above the varying behaviour of dim M ∩ S_{n}(v) is exhibited for values of v bounded away from the Dirichlet exponent 1/n. Nevertheless, it is believed that when v lies in a critical range near the Dirichlet exponent 1/n then, for a wide class of manifolds (including non–degenerate manifolds), the behaviour of dim M∩S_{n}(v) can be captured by a single, general formula. That is to say, that dim M∩S_{n}(v) is independent of the arithmetic properties of M for v close to 1/n. We shall prove that this is indeed the case for planar curves. Note that for planar curves the Dirichlet exponent is 1/2 and that the above ‘circles example’ shows that any critical range for v is a subset of [1/2, 1]. In general, the critical range is governed by the dimension of the ambient space and the dimension of the manifold.

Before stating our results we introduce the notion of lower order. Given an approximating function ψ, the lower order λ_{ψ} of 1/ψ is defined by

λ_{ψ} := lim inf_{h→∞} \frac{− log ψ(h)}{log h} ,

and indicates the growth of the function 1/ψ ‘near’ infinity. Note that λ_{ψ} is non-negative since ψ is a decreasing function. Regarding Problem 2, our main results are as follows.

Theorem 3 Let f ∈ C^{(3)}(I_{0}), where I_{0} is an interval and C_{f} := {(x, f(x)) : x ∈ I_{0}}. Assume that there exists at least one point on the curve C_{f} which is non-degenerate. Let s ∈ (1/2, 1) and ψ be an approximating function. Then

H^{s}(C_{f} ∩ S_{2}(ψ)) = ∞ if ∑_{h=1}^{∞} h^{1−s} ψ(h)^{s+1} = ∞ .

Theorem 4 Let f ∈ C^{(3)}(I_{0}), where I_{0} is an interval and C_{f} := {(x, f(x)) : x ∈ I_{0}}. Let ψ be an approximating function with λ_{ψ} ∈ [1/2, 1). Assume that

dim \{x ∈ I_{0} : f′′(x) = 0\} ⩽ \frac{2 − λ_{ψ}}{1 + λ_{ψ}} .  (1)

Then

dim C_{f} ∩ S_{2}(ψ) = d := \frac{2 − λ_{ψ}}{1 + λ_{ψ}} .

Furthermore, suppose that λ_{ψ} ∈ (1/2, 1). Then

H^{d}(C_{f} ∩ S_{2}(ψ)) = ∞ if lim sup_{h→∞} h^{2−s}ψ(h)^{s+1} > 0 .

By considering the function ψ : h → h^{−v}, an immediate consequence of the theorems is the following corollary.

Corollary 2 Let f ∈ C^{(3)}(I_{0}), where I_{o} is an interval and C_{f} := {(x, f(x)) : x ∈ I_{0}}. Let v ∈ [1/2, 1) and assume that dim {x ∈ I_{0} : f′′(x) = 0} ⩽ (2 − v)/(1 + v). Then

dim C_{f} ∩ S_{2}(v) = d := \frac{2 − v}{1 + v} .

Moreover, if v ∈ (1/2, 1) then H^{d}(C_{f} ∩ S_{2}(v)) = ∞.

Remark. Regarding Theorem 4, the hypothesis (1) on the set {x ∈ I_{0} : f′′(x) = 0} is stronger than simply assuming that the curve C_{f} is non-degenerate. It requires the curve to be non– degenerate everywhere except on a set of Hausdorff dimension no larger than (2−λ_{ψ})/(1+λ_{ψ})

– rather than just measure zero. Note that the hypothesis can be made independent of the lower order λ_{ψ} (or indeed of v in the case of the corollary) by assuming that dim{x ∈ I_{0} : f′′(x) = 0} ≤ 1/2. The proof of Theorem 4 follows on establishing the upper and lower bounds for dim C_{f} ∩ S_{2}(ψ) separately. Regarding the lower bound statement, all that is required is that there exists at least one point on the curve C_{f} which is non-degenerate. This is not at all surprising since the lower bound statement can be viewed as a simple consequence of Theorem 3. The hypothesis (1) is required to obtain the upper bound dimension statement. Even for non-degenerate curves, without such a hypothesis the statement of Theorem 4 is clearly false as the following example shows.

Example: The Cantor curve. Let K denote the standard middle third Cantor set obtained by removing the middle third of the unit interval [0, 1] and then inductively repeating the process on each of the remaining intervals. For our purpose, a convenient expression for K is the following:

⋂_{i=1}^{∞}([0, 1] \ ⋃_{j=1}^{2^{i−1}} I_{i,j}) = [0, 1] \ ⋃_{i=1}^{∞} ⋃_{j=1}^{2^{i−1}} I_{i,j} ,

where I_{i,j} is the j^{th} interval of the 2^{i−1} open intervals of length 3^{−i} removed at the i^{th}-level of the Cantor construction. Note that the intervals I_{i,j} are pair wise disjoint. Give a pair (i, j), define the function

f_{i,j} : x → f_{i,j}(x) := \{ \begin{matrix}e^{−i − \frac{1}{(x−a)(b−x)}} & if x ∈ I_{i,j} \\ 0 & if x ∈ [0, 1] \ I_{i,j} \end{matrix} ,

where a and b are the end points of the interval I_{i,j}. Now set

f : x → f(x) := ∑_{i=1}^{∞} ∑_{j=1}^{2^{i−1}} f_{i,j}(x) .

Note that the function f is obviously C^{(∞)} as the sum converges uniformly. Also, for x ∈ K and m ∈ ℕ we have that f_{i,j}^{(m)}(x) = 0 and so

f^{(m)}(x) = ∑_{i=1}^{∞} ∑_{j=1}^{2^{i−1}} f_{i,j}^{(m)}(x) = 0 .

On the other hand, for x ∈ [0, 1] ∖ K we have that f^{(m)}(x) > 0. Thus the curve C_{K} = {(x, f(x)) : x ∈ (0, 1)} is exactly degenerate on K and non-degenerate elsewhere. Note that C_{K} is a non-degenerate curve since K is of Lebesgue measure zero. The upshot of this is that for any x ∈ K the point (x, f(x)) is 1-approximable; i.e. there exists infinitely many q ∈ ℕ such that

∥qx∥ < q^{−1}  and  ∥qf(x)∥ < q^{−1} .

The second inequality is trivial as f(x) = 0 and the first inequality is a consequence of Dirichlet’s theorem. Thus,

dim C_{K} ∩ S_{2}(v) ≥ dim K = log 2/ log 3

irrespective of v ∈ (1/2, 1). Obviously, by choosing Cantor sets K with dimension close to one, we can ensure that dim C_{K} ∩ S_{2}(v) is close to one irrespective of v ∈ (1/2, 1).

For simultaneous Diophantine approximation on planar curves, Theorem 3 is the precise analogue of the divergent part of Jarník’s theorem and Theorem 4 establishes a complete Hausdorff dimension theory.

Note that the measure part of Theorem 4 is substantially weaker than Theorem 3 – the general measure statement. For example, with v ∈ (1/2, 1) and α = 1/(d + 1) consider the approximating function ψ given by

ψ : h → h^{−v}(log h)^{−α} .

Then λ_{ψ} = v and assuming that (1) is satisfied, the dimension part of Theorem 4 implies that

dim C_{f} ∩ S_{2}(ψ) = d := \frac{2 − v}{1 + v} .

However,

lim sup_{h→∞} h^{2−d}ψ(h)^{d+1} = lim_{h→∞}(log h)^{−1} = 0

and so the measure part of Theorem 4 is not applicable. Nevertheless,

∑ h^{1−d}ψ(h)^{d+1} = ∑(h log h)^{−1} = ∞

and Theorem 3 implies that H^{d}(C_{f} ∩ S_{2}(ψ)) = ∞ .

Theorem 3 falls short of establishing a complete Hausdorff measure theory for simultaneous Diophantine approximation on planar curves. In its simplest form, it should be possible to summarize the Hausdorff measure theory by a clear cut statement of the following type.

Conjecture H Let s ∈ (1/2, 1) and ψ be an approximating function. Let f ∈ C^{(3)}(I_{0}), where I_{o} is an interval and C_{f} := {(x, f(x)) : x ∈ I_{0}}. Assume that dim{x ∈ I_{0} : f′′(x) = 0} ≤ 1/2. Then

H^{s} (C_{f} ∩ S_{2}(ψ)) = \{ \begin{matrix}0 & if ∑ h^{1−s} ψ(h)^{s+1} < ∞ \\ ∞ & if ∑ h^{1−s} ψ(h)^{s+1} = ∞ \end{matrix} .

The divergent part of the above statement is Theorem 3. As with the ‘Khintchine theory’, the above convergent part would follow on proving Conjecture 1 of §1.4. However, for rational quadrics we are able to prove the convergent result independently of any conjecture.

Theorem 5 Let s ∈ (1/2, 1) and ψ be an approximating function. Then for any nondegenerate, rational quadric Q we have that

H^{s} (Q ∩ S_{2}(ψ)) = 0 if ∑ h^{1−s} ψ(h)^{s+1} < ∞ .

1.4 Rational points close to a curve

First some useful notation. For any point r ∈ ℚ^{n} there exists the smallest q ∈ ℕ such that qr ∈ ℤ^{n}. Thus, every point r ∈ ℚ^{n} has a unique representation in the form

\frac{p}{q} = \frac{(p_{1}, …, p_{n})}{q} = (\frac{p_{1}}{q} , …, \frac{p_{n}}{q} )

with (p_{1}, …, p_{n}) ∈ ℤ^{n}. Henceforth, we will only consider points of ℚ^{n} in this form.

Understanding the distribution of rational points close to a reasonably defined curve is absolutely crucial towards making any progress with the main problems considered in this paper. More precisely, the behaviour of the following counting function will play a central role. The function N_{f}(Q, ψ, I). Let I_{0} denote a finite, open interval of ℝ and let f be a function in C^{(3)}(I_{0}) such that

0 < c_{1} := inf_{x∈I_{0}} |f′′(x)| ≤ c_{2} := sup_{x∈I_{0}} |f′′(x)| < ∞ .  (2)

Given an interval I ⊆ I_{0}, an approximating function ψ and Q ∈ ℝ^{+} consider the counting function N_{f}(Q, ψ, I) given by

N_{f}(Q, ψ, I) := #{p/q ∈ ℚ^{2} : q ⩽ Q, p_{1}/q ∈ I, |f(p_{1}/q) − p_{2}/q| < ψ(Q)/Q}.

In short, the function N_{f}(Q, ψ, I) counts ‘locally’ the number of rational points with bounded denominator lying within a specified neighbourhood of the curve parameterized by f. In [20], Huxley obtains a reasonably sharp upper bound for N_{f}(Q, ψ, I). We will obtain an exact lower bound and also prove that the rational points under consideration are ‘evenly’ distributed. The proofs of the Khintchine type and Hausdorff measure/dimension theorems stated in this paper rely heavily on this information. In particular, the exact upper bound in Theorem 4 is easily established in view of Huxley’s result [20, Theorem 4.2.4] which we state in a simplified form. Huxley’s estimate: Let ψ be an approximating function such that tψ(t) → ∞ as t → ∞. For ε > 0 and Q sufficiently large

N_{f}(Q, ψ, I_{0}) ⩽ ψ(Q) Q^{2+ε}.  (3)

The complementary lower bound is the substance of our next result.

Theorem 6 Let ψ be an approximating function satisfying

lim_{t→+∞} ψ(t) = lim_{t→+∞} \frac{1}{tψ(t)} = 0.  (4)

There exists a constant c > 0, depending on I, such that for Q sufficiently large

N_{f}(Q, ψ, I) ⩾ c Q^{2} ψ(Q) |I| .

We suspect that the lower bound given by Theorem 6 is best possible up to a constant multiple. It is plausible that for compact curves, the constant c is independent of I.

Regarding Huxley’s estimate, the presence of the ‘ε’ factor prevents us from proving the desired ‘convergent’ measure theoretic results. We suspect that a result of the following type is in fact true – proving it is another matter.

Conjecture 1 Let ψ be an approximating function such that tψ(t) → ∞ as t → ∞. There exists a constant ĉ > 0 such that for Q sufficiently large

N_{f}(Q, ψ, I_{0}) ⩽ ĉQ^{2} ψ(Q) .

Conjecture 1 has immediate consequences for the main problems considered in this paper. In particular, it would imply the following.

Conjecture 2 Any C^{(3)} non–degenerate planar curve is of Khintchine type for convergence.

Conjecture 2 would naturally complement Theorem 1 of this paper. The implication Conjecture 1 ⟹ Conjecture 2 is reasonably straightforward – simply modify the argument set out in the proof of Theorem 2. Also, it is not difficult to verify that Conjecture 1 implies the ‘convergent’ part of Conjecture H – simply modify the argument set out in the proof of Theorem 5. An intriguing problem is to determine whether or not the two conjectures stated above are in fact equivalent.

2. Proof of the rational quadric statements

2.1 Proof of Theorem 2

The divergence part of the theorem is a trivial consequence of Corollary 1 to Theorem 1. To establish the convergence part we proceed as follows.

Let ψ be an approximating function such that ∑ ψ(h)^{2} < ∞. The claim is that |Q(ψ)|_{Q} = 0. We begin by introducing an auxiliary function Ψ given by

Ψ(h) := max \{ψ(h), h^{− \frac{1}{2}} (log h)^{−1}\} .

Clearly, Ψ is an approximating function and furthermore

∑ Ψ(h)^{2} < ∞  and  Ψ(h) ≥ ψ(h) .

Thus Q(ψ) ⊂ Q(Ψ) and the claim will follow on showing that |Q(Ψ)|_{Q} = 0. It is easily verified that such a ‘zero’ statement is invariant under rational affine transformations of the plane. In view of this, it suffices to consider the curves C_{1}, C_{1}^{∗} and V_{2} – see §1.2.1.

In the following, C(q; s, t) will denote the square with centre at the rational point (s/q, t/q) and of side length 2Ψ(q)/q. Case (a): Q = C_{1}. For m ∈ ℕ, let

W_{m}(Ψ; C_{1}) := ⋃_{2^{m}<q≤2^{m+1}} ⋃_{(s,t)∈ℤ^{2}} C_{1} ∩ C(q; s, t) .

Then C_{1}(Ψ) = lim sup_{m→∞} W_{m}(Ψ; C_{1}) and in view of the Borel-Cantelli lemma |C_{1}(Ψ)|_{C_{1}} = 0 if ∑ |W_{m}(Ψ; C_{1})|_{C_{1}} < ∞. Next, note that if C_{1}∩C(q; s, t) ≠ ∅ then (q−2\sqrt{2}Ψ(q))^{2} ≤ s^{2}+t^{2} ≤ (q + 2\sqrt{2}Ψ(q))^{2} and |C_{1} ∩ C(q; s, t)|_{C_{1}} ≪ Ψ(q)/q. It follows that

|W_{m}(Ψ; C_{1})|_{C_{1}} ≪ ∑_{2^{m}<q≤2^{m+1}} ∑_{\begin{matrix}(s,t)∈ℤ^{2}: \\ (q−2\sqrt{2}Ψ(q))^{2} ≤s^{2}+t^{2} ≤ (q+2\sqrt{2}Ψ(q))^{2} \end{matrix}} |C_{1} ∩ C(q; s, t)|_{C_{1}}

≪ \frac{Ψ(2^{m})}{2^{m}} ∑_{2^{m}<q≤2^{m+1}} ∑_{\begin{matrix}n: \\ |q−\sqrt{n}|<4Ψ(q) \end{matrix}} r(n) ,  (5)

where r(n) denotes the number of representations of n as the sum of two squares.

With reference to Theorem A of Appendix II, with ψ := 4Ψ, Q := 2^{m} and N := [Q/Ψ(Q)] it is easily verified that the error term associated with ∑_{Q<q⩽2Q} _{∑n}′ r(n) is

≪ Q^{\frac{15}{8}} (log Q)^{65}Ψ(Q) .

Here we use the trivial fact that Ψ(Q^{∗}) := Ψ(Q + 1) ≤ Ψ(Q) since Ψ is decreasing. On the other hand, for the main term we have that

Q^{2}Ψ(2Q) ≪ ∑_{Q<q⩽2Q} qΨ(q) ≪ Q^{2}Ψ(Q) .

Thus, Theorem A implies that

∑_{2^{m}<q≤2^{m+1}} ∑_{\begin{matrix}n: \\ |q−\sqrt{n}|<4Ψ(q) \end{matrix}} r(n) ≪ 2^{2m} Ψ(2^{m}) .  (6)

This estimate together with (5) implies that |W_{m}(Ψ; C_{1})|_{C_{1}} ≪ 2^{m} Ψ(2^{m})^{2}. In turn, we obtain that

∑_{m∈ℕ} |W_{m}(Ψ; C_{1})|_{C_{1}} ≪ ∑_{m∈ℕ}2^{m} Ψ(2^{m})^{2} ≍ ∑_{h∈ℕ}Ψ(h)^{2} < ∞ .

This completes the proof of the theorem in the case that Q is the image of the unit circle C_{1} under a rational affine transformation of the plane. The other two cases are similar. The key is to bring (6) into play.

Case (b): Q = C_{1}^{∗}. For k ∈ ℕ, let C_{1;k}^{∗} := {(x_{1}, x_{2}) ∈ ℝ^{2} : x_{1}^{2} − x_{2}^{2} = 1 with |x_{1}| ≤ 2^{k}}. Thus, C_{1;k}^{∗} is the hyperbola C_{1}^{∗} with the first co-ordinate bounded above by 2^{k}. For m ∈ ℕ, let

W_{m}(Ψ; C_{1;k}^{∗}) := ⋃_{2^{m}<q≤2^{m+1}} ⋃_{(s,t)∈ℤ^{2}} C_{1;k}^{∗} ∩ C(q; s, t)

and let C_{1;k}^{∗}(Ψ) := lim sup_{m→∞} W_{m}(Ψ; C_{1;k}^{∗}). Clearly, C_{1}^{∗}(Ψ) = ⋃_{k=1}^{∞} C_{1;k}^{∗}(Ψ) and so |C_{1}^{∗}(Ψ)|_{C_{1}^{∗}} = 0 if |C_{1;k}^{∗}(Ψ)|_{C_{1}^{∗}} = 0 for each k ∈ ℕ. The latter follows on showing that ∑ |W_{m}(Ψ; C_{1;k}^{∗})|_{C_{1}^{∗}} < ∞.

It is easily verified that if C_{1;k}^{∗} ∩ C(q; s, t) ≠ ∅ then 1/2 < |s|/q < a := 2^{k+1}, |t| < |s| and

|q^{2} + t^{2} − s^{2}| < 8 |s| Ψ(q) + 8 Ψ(q)^{2} < 8 |s| Ψ(|s|/a) + 8 Ψ(|s|/a)^{2} .

Here we have used that fact that the function Ψ is decreasing. It follows via (6), that for m sufficiently large

|W_{m}(Ψ; C_{1;k}^{∗})|_{C_{1}} ≪ \frac{Ψ(2^{m})}{2^{m}} ∑_{2^{m}<q≤2^{m+1}} ∑_{\begin{matrix}(s,t)∈ℤ^{2} : q/2<s<aq \\ (s−8Ψ(s/a))^{2} ≤q^{2}+t^{2} ≤ (s+8Ψ(s/a))^{2} \end{matrix}} 1

≤ \frac{Ψ(2^{m})}{2^{m}} ∑_{2^{m−1}<s≤a2^{m+1}}  ∑_{\begin{matrix}n: \\ |s−\sqrt{n}|<8Ψ(s/a) \end{matrix}} r(n)

≤ \frac{Ψ(2^{m})}{2^{m}} ∑_{i=0}^{k+2}  ∑_{2^{m+i−1}<s≤2^{m+i}}  ∑_{\begin{matrix}n: \\ |s−\sqrt{n}|<8Ψ(s/a) \end{matrix}} r(n)

≪ k \frac{Ψ(2^{m})}{2^{m}} 2^{2(m+k+1)} Ψ(2^{m−k−2}) ≪ k 2^{3k} 2^{m−k−2} Ψ(2^{m−k−2})^{2} .

Thus, ∑ |W_{m}(Ψ; C_{1;k}^{∗})|_{C_{1}^{∗}} ≪ ∑ 2^{m} Ψ(2^{m})^{2} ≍ ∑ Ψ(h)^{2} < ∞ and we are done.

Case (c): Q = V_{2}. For k ∈ ℕ, let V_{2;k} := {(x_{1}, x_{2}) ∈ ℝ^{2} : x_{2} = x_{1}^{2} with |x_{1}| ≤ 2^{k}}. For m ∈ ℕ, let

W_{m}(Ψ; V_{2;k}) := ⋃_{2^{m}<q≤2^{m+1}} ⋃_{(s,t)∈ℤ^{2}} V_{2;k} ∩ C(q; s, t) .

We need to show that ∑ |W_{m}(Ψ; V_{2;k})|_{V_{2}} < ∞. It is easily verified that if V_{2;k} ∩ C(q; s, t) ≠ ∅ then 0 ≤ |s|/q < a := 2^{k+1}, −1 < t/q < a^{2} and |s^{2} − tq| < 2 Ψ(q)(2 |s| + |t|) + 4 Ψ(q)^{2} < 6 a^{2}qΨ(q) + 4 Ψ(q)^{2}; that is,

|(2s)^{2} − 4tq| < 24 a^{2}qΨ(q) + 16 Ψ(q)^{2} .  (7)

Let w := q + t and z := q − t. Then, 2q = w + z, 2t = w − z and q − 1 < w < q(a^{2} + 1). Furthermore, (7) becomes

|(2s)^{2} + z^{2} − w^{2}| < 24 a^{2}qΨ(q) + 16 Ψ(q)^{2} < 48 a^{2} w Ψ (\frac{w}{(a^{2}+1)}) + 16 Ψ (\frac{w}{(a^{2}+1)})^{2} . (8)

It follows, that for m sufficiently large

|W_{m}(Ψ; V_{2;k})|_{V_{2}} ≪ \frac{Ψ(2^{m})}{2^{m}} ∑_{2^{m}<q≤2^{m+1}}  ∑_{\begin{matrix}(s,t)∈ℤ^{2} : −q<t<a^{2}q \\ (7) holds \end{matrix}} 1

≤ \frac{Ψ(2^{m})}{2^{m}} ∑_{2^{m−1}<w≤(a^{2}+1)2^{m+1}}  ∑_{(s,z)∈ℤ^{2} : (8) holds} 1

≤ \frac{Ψ(2^{m})}{2^{m}} ∑_{2^{m−1}<w≤a^{2}2^{m+2}}  ∑_{\begin{matrix}n: \\ |w−\sqrt{n}|<48Ψ(w/(2a^{2})) \end{matrix}} r(n) .

As in case (b), the desired statement now follows on using (6) to estimate the double sum. ♠

Before moving onto the proof of Theorem 5, we define Hausdorff measure and dimension for the sake of completeness and in order to establish some notation.

2.2 Hausdorff measure and dimension

The Hausdorff dimension of a non–empty subset X of n–dimensional Euclidean space ℝ^{n}, is an aspect of the size of X that can discriminate between sets of Lebesgue measure zero.

For ρ > 0, a countable collection {C_{i}} of Euclidean cubes in ℝ^{n} with side length l(C_{i}) ≤ ρ for each i such that X ⊂ ⋃_{i} C_{i} is called a ρ-cover for X. Let s be a non-negative number and define

H_{ρ}^{s}(X) = inf {∑_{i} l^{s}(C_{i}) : {C_{i}} is a ρ−cover of X} ,

where the infimum is taken over all possible ρ-covers of X. The s-dimensional Hausdorff measure H^{s}(X) of X is defined by

H^{s}(X) = lim_{ρ→0} H_{ρ}^{s}(X) = sup_{ρ>0} H_{ρ}^{s}(X)

and the Hausdorff dimension dim X of X by

dim X = inf {s : H^{s}(X) = 0} = sup {s : H^{s}(X) = ∞} .

Strictly speaking, in the standard definition of Hausdorff measure the ρ–cover by cubes is replaced by non–empty subsets in ℝ^{n} with diameter at most ρ. It is easy to check that the resulting measure is comparable to H^{s} defined above and thus the Hausdorff dimension is the same in both cases. For our purpose using cubes is just more convenient. Moreover, if H^{s} is zero or infinity then there is no loss of generality by restricting to cubes. Further details and alternative definitions of Hausdorff measure and dimension can be found in [19, 25].

2.3 Proof of Theorem 5

To a certain degree the proof follows the same line of argument as the proof of the convergent part of Theorem 2. In particular, it suffices to consider the rational quadrics C_{1}, C_{1}^{∗} and V_{2}. Below, we consider the case of the unit circle C_{1} and leave the hyperbola C_{1}^{∗} and parabola V_{2} to the reader. The required modifications are obvious.

Let ψ be an approximating function such that ∑ h^{1−s} ψ(h)^{s+1} < ∞ and consider the auxiliary function Ψ given by

Ψ(h) := max {ψ(h), h^{−1} (log h)^{260}} .

Clearly, Ψ is an approximating function and since s > 1/2 we have that ∑ h^{1−s} Ψ(h)^{s+1} < ∞. With the same notation as in the proof of Theorem 2, for each l ∈ ℕ

{W_{m}(Ψ, C_{1}) : m = l, l + 1, … }

is a cover for C_{1}(Ψ) := C_{1} ∩ S_{2}(ψ) by squares C(q; s, t) of maximal side length 2Ψ(2^{l})/2^{l}. It follows from the definition of s–dimensional Hausdorff measure that with ρ := 2Ψ(2^{l})/2^{l}

H_{ρ}^{s}(C_{1}(Ψ)) ≤ ∑_{m=l}^{∞} ∑_{2^{m}<q≤2^{m+1}} ∑_{\begin{matrix}(s,t)∈ℤ^{2}: \\ (q−2\sqrt{2}Ψ(q))^{2} ≤s^{2}+t^{2} (q+2\sqrt{2}Ψ(q))^{2} \end{matrix}} (\frac{2Ψ(2^{m})}{2^{m}})^{s}

≪ ∑_{m=l}^{∞} (\frac{Ψ(2^{m})}{2^{m}})^{s} ∑_{2^{m}<q≤2^{m+1}} ∑_{\begin{matrix}n: \\ |q−\sqrt{n}|<4Ψ(q) \end{matrix}} r(n) .

In view of Theorem A of Appendix II, the contribution from the two inner sums is ≪ 2^{2m}Ψ(2^{m}). Thus,

H_{ρ}^{s}(C_{1}(Ψ)) ≪ ∑_{m=l}^{∞} 2^{m(2−s)} Ψ(2^{m})^{1+s} → 0

as ρ → 0; or equivalently at l → ∞. Hence, H^{s}(C_{1}(ψ)) ≤ H^{s}(C_{1}(Ψ)) = 0 as required. ♠

3. Ubiquitous systems

In [6], a general framework is developed for establishing divergent results analogous to those of Khintchine and Jarník (see §1.1) for a natural class of lim sup sets. The framework is based on the notion of ‘ubiquity’, which goes back to [2] and [16] and captures the key measure theoretic structure necessary to prove such measure theoretic laws. The ‘ubiquity’ introduced below is a much simplified version of that in [6] and takes into consideration the specific applications that we have in mind.

3.1 Ubiquitous systems in ℝ

Let I_{0} be an interval in ℝ and R := (R_{α})_{α∈J} be a family of resonant points R_{α} of I_{0} indexed by an infinite, countable set J. Next let β : J → ℝ^{+} : α ↦ β_{α} be a positive function on J. Thus, the function β attaches a ‘weight’ β_{α} to the resonant point R_{α}. Also, for t ∈ ℕ let J(t) := {α ∈ J : β_{α} ⩽ 2^{t}} and assume that #J(t) is always finite. Given an approximating function Ψ let

Λ(R, β, Ψ) := {x ∈ I_{0} : |x − R_{α}| < Ψ(β_{α}) for infinitely many α ∈ J } .

The set Λ(R, β, Ψ) is easily seen to be a lim sup set. The general theory of ubiquitous systems developed in [6], provides a natural measure theoretic condition for establishing divergent results analogous to those of Khintchine and Jarník for Λ(R, β, Ψ). Since Λ(R, β, Ψ) is a subset of I_{0}, any Khintchine type result would naturally be with respect to one-dimensional Lebesgue measure | . |.

Throughout, ρ : ℝ^{+} → ℝ^{+} will denote a function satisfying lim_{t→∞} ρ(t) = 0 and is usually referred to as the ubiquitous function. Also B(x, r) will denote the ball (or rather the interval) centred at x or radius r.

Definition 2 (Ubiquitous systems on the real line) Suppose there exists a function ρ and an absolute constant κ > 0 such that for any interval I ⊆ I_{0}

lim inf_{t→∞} |_{⋃α∈J(t)} (B(R_{α}, ρ(2^{t})) ∩ I)| ⩾ κ |I| .

Then the system (R; β) is called locally ubiquitous in I_{0} with respect to ρ.

The consequences of this definition of ubiquity are the following key results.

Lemma 1 Suppose that (R, β) is a local ubiquitous system in I_{0} with respect to ρ and let Ψ be an approximating function such that Ψ(2^{t+1}) ⩽ \frac{1}{2}Ψ(2^{t}) for t sufficiently large. Then

|Λ(R, β, Ψ)| = FULL := |I_{0}|  if  ∑_{t=1}^{∞} \frac{Ψ(2^{t})}{ρ(2^{t})} = ∞ .

Lemma 2 Suppose that (R, β) is a local ubiquitous system in I_{0} with respect to ρ and let Ψ be an approximating function. Let s ∈ (0, 1) and let

G := lim sup_{t→∞} \frac{Ψ(2^{t})^{s}}{ρ(2^{t})} .

(i) Suppose that G = 0 and that Ψ(2^{t+1}) ⩽ \frac{1}{2}Ψ(2^{t}) for t sufficiently large. Then,

H^{s}(Λ(R, β, Ψ)) = ∞  if  ∑_{t=1}^{∞} \frac{Ψ(2^{t})^{s}}{ρ(2^{t})} = ∞ .

(ii) Suppose that G > 0. Then, H^{s}(Λ(R, β, Ψ)) = ∞.

Corollary 3 Suppose that (R, β) is a local ubiquitous system in I_{0} with respect to ρ and let Ψ be an approximating function. Then

dim(Λ(R, β, Ψ)) ⩾ d := min \{1, |lim sup_{t→∞} \frac{log ρ(2^{t})}{log Ψ(2^{t})}|\} .

Moreover, if d < 1 and lim sup_{t→∞} Ψ(2^{t})^{d}/ρ(2^{t}) > 0, then H^{d}(Λ(R, β, Ψ)) = ∞.

The concept of ubiquity was originally formulated by Dodson, Rynne & Vickers [16] to obtain lower bounds for the Hausdorff dimension of lim sup sets. In the one-dimensional setting considered here, their ‘ubiquity result’ essentially corresponds to Corollary 3 above. Furthermore, the ubiquitous systems of [16] essentially coincide with the regular systems of Baker & Schmidt [2] and both have proved very useful in obtaining lower bounds for the Hausdorff dimension of lim sup sets. However, unlike the framework developed in [6], both [2] and [16] fail to shed any light on establishing the more desirable divergent Khintchine and Jarník type results. The latter, clearly implies lower bounds for the Hausdorff dimension. For further details regarding regular systems and the original formulation of ubiquitous systems see [6, 11].

Lemmas 1 and 2 follow directly from Corollaries 2 and 4 in [6]. Note that in Lemma 2, if G > 0 then the divergent sum condition of part (i) is trivially satisfied. The dimension statement (Corollary 3) is a consequence of part (ii) of Lemma 2 and so the regularity condition 2 Ψ(2^{t+1}) ⩽ Ψ(2^{t}) on the function Ψ is not necessary – see [6, Corollary 6].

The framework and results of [6] are abstract and general unlike the concrete situation described above. In view of this and for the sake of completeness we retraced the argument of [6] in the above simple setting at the end of the paper §A-C. This has the effect of making the paper self-contained and more importantly should help the interested reader with understanding the abstract approach undertaken in [6]. The direct proofs of Lemmas 1 and 2 are substantially easier (both technically and conceptionally) than the general statements of [6].

3.2 Ubiquitous systems close to a curve in ℝ^{n}

In this section we develop the theory of ubiquity to incorporate the situation in which the resonant points of interest lie within some specified neighborhood of a given curve in ℝ^{n}.

With n ≥ 2, let R := (R_{α})_{α∈J} be a family of resonant points R_{α} of ℝ^{n} indexed by an infinite set J. As before, β : J → ℝ^{+} : α ↦ β_{α} is a positive function on J. For a point R_{α} in R, let R_{α,k} represent the k’th coordinate of R_{α}. Thus, R_{α} := (R_{α,1}, R_{α,2}, …, R_{α,n}). Throughout this section and the remainder of the paper we will use the notation R_{C}(Φ) to denote the sub-family of resonant points R_{α} in R which are “Φ–close” to the curve C = C_{f} := {(x, f_{2}(x), …, f_{n}(x)) : x ∈ I_{0}} where Φ is an approximating function, f = (f_{1}, …, f_{n}) : I_{0} → ℝ^{n} is a continuous map with f_{1}(x) = x and I_{0} is an interval in ℝ. Formally, and more precisely

R_{C}(Φ) := (R_{α})_{α∈J_{C}(Φ)} where J_{C}(Φ) := {α ∈ J : max_{1⩽k⩽n} |f_{k}(R_{α,1}) − R_{α,k}| < Φ(β_{α})} .

Finally, we will denote by R_{1} the family of first co-ordinates of the points in R_{C}(Φ); that is

R_{1} := (R_{α,1})_{α∈J_{C}(Φ)} .

By definition, R_{1} is a subset of the interval I_{0} and can therefore be regarded as a set of resonant points for the theory of ubiquitous systems in ℝ. This leads us naturally to the following definition in which the ubiquity function ρ is as in §3.1.

Definition 3 (Ubiquitous systems near curves) The system (R_{C}(Φ), β) is called locally ubiquitous with respect to ρ if the system (R_{1}, β) is locally ubiquitous in I_{0} with respect to ρ.

Next, given an approximating function Ψ let Λ(R_{C}(Φ), β, Ψ) denote the the set x ∈ I_{0} for which the system of inequalities

\{ \begin{matrix}|x − R_{α,1}| & < & Ψ(β_{α}) \\ max_{2⩽k⩽n} |f_{k}(x) − R_{α,k}| & < & Ψ(β_{α}) + Φ(β_{α}) , \end{matrix}

is simultaneously satisfied for infinitely many α ∈ J. The following two lemmas are the analogues of Lemmas 1 and 2 for the case of ubiquitous systems close to a curve. Similarly, Corollary 4 is the analogue of Corollary 3.

Lemma 3 Consider the curve C := {(x, f_{2}(x), …, f_{n}(x)) : x ∈ I_{0}}, where f_{2}, …, f_{n} are locally Lipshitz in a finite interval I_{0}. Let Φ and Ψ be approximating functions. Suppose that (R_{C}(Φ), β) is a locally ubiquitous system with respect to ρ. If Ψ and ρ satisfy the conditions of Lemma 1 then

| Λ (R_{C}(Φ), β, Ψ) | = |I_{0}| .

Lemma 4 Consider the curve C := {(x, f_{2}(x), …, f_{n}(x)) : x ∈ I_{0}}, where f_{2}, …, f_{n} are locally Lipshitz in a finite interval I_{0}. Let Φ and Ψ be approximating functions. Suppose that (R_{C}(Φ), β) is a locally ubiquitous system with respect to ρ. Let s ∈ (0, 1) and let

G := lim sup_{t→∞} \frac{Ψ(2^{t})^{s}}{ρ(2^{t})} .

(i) Suppose that G = 0 and that Ψ(2^{t+1}) ⩽ \frac{1}{2}Ψ(2^{t}) for t sufficiently large. Then,

H^{s}(Λ(R_{C}(Φ), β, Ψ)) = ∞  if  ∑_{t=1}^{∞} \frac{Ψ(2^{t})^{s}}{ρ(2^{t})} = ∞ .

(ii) Suppose that G > 0. Then, H^{s}(Λ(R_{C}(Φ), β, Ψ)) = ∞.

Corollary 4 Consider the curve C := {(x, f_{2}(x), …, f_{n}(x)) : x ∈ I_{0}}, where f_{2}, …, f_{n} are locally Lipshitz in a finite interval I_{0}. Let Φ and Ψ be approximating functions. Suppose that (R_{C}(Φ), β) is a locally ubiquitous system with respect to ρ. Then

dim Λ (R_{C}(Φ), β, Ψ) ⩾ d := min \{1, |lim sup_{t→∞} \frac{log ρ(2^{t})}{log Ψ(2^{t})}|\} .

Moreover, if d < 1 and lim sup_{t→∞} Ψ(2^{t})^{d}/ρ(2^{t}) > 0, then H^{d}(Λ (R_{C}(Φ), β, Ψ)) = ∞.

Proof of Lemmas 3 & 4 and Corollary 4. It suffices to prove the lemmas for a sufficiently small neighborhood of a fixed point in I_{0}. Therefore, there is no loss of generality in assuming that f_{2}, …, f_{n} satisfy the Lipshitz condition on I_{0}. Thus, we can fix a constant c_{3} ⩾ 1 such that for k ∈ {2, …, n} and x, y ∈ I_{0}

|f_{k}(x) − f_{k}(y)| ⩽ c_{3}|x − y|.  (9)

Since (R_{C}(Φ), β) is a locally ubiquitous system with respect to ρ, by definition (R_{1}, β) is a locally ubiquitous system in I_{0} with respect to ρ. The set Λ(R_{1}, β, Ψ/c_{3}) consists of x ∈ I_{0} for which the inequality

|x − R_{α,1}| < Ψ(β_{α})/c_{3} ⩽ Ψ(β_{α})  (10)

is satisfied for infinitely many α ∈ J_{C}(Φ). Suppose x satisfies (10) for some α ∈ J_{C}(Φ). In view of (9), |f_{k}(x) − f_{k}(R_{α,1})| ⩽ c_{3}|x − R_{α,1}| which implies that

|f_{k}(x) − R_{α,k}| = |f_{k}(x) − f_{k}(R_{α,1}) + f_{k}(R_{α,1}) − R_{α,k}|
⩽ |f_{k}(x) − f_{k}(R_{α,1})| + |f_{k}(R_{α,1}) − R_{α,k}|
⩽ c_{3}|x − R_{α,1}| + Φ(β_{α})
< c_{3} · Ψ(β_{α})/c_{3} + Φ(β_{α}) = Ψ(β_{α}) + Φ(β_{α}).

Thus Λ(R_{1}, β, Ψ/c_{3}) ⊂ Λ(R, β, Ψ). Applying Lemmas 1 & 2 and Corollary 3 to the set Λ(R_{1}, β, Ψ/c_{3}) gives the desired statements concerning the set Λ(R_{C}(Φ), β, Ψ). ♠

4. Proof of Theorem 6

We begin by stating a key result which not only implies Theorem 6 but gives rise to a ubiquitous system that will be required in proving Theorems 1 and 4.

4.1 The ubiquity version of Theorem 6

Theorem 7 Let I_{0} denote a finite, open interval of ℝ and let f be a function in C^{(3)}(I_{0}) satisfying (2). Let ψ be an approximating function satisfing (4). Then for any interval I ⊆ I_{0} there exist constants δ_{0}, C_{1} > 0 such that for Q sufficiently large

| ⋃_{p/q∈A_{Q}(I)} (B (\frac{p_{1}}{q} , \frac{C_{1}}{Q^{2}ψ(Q)}) ∩ I)| ⩾ \frac{1}{2} |I| ,

where

A_{Q}(I) := {p/q ∈ ℚ^{2} : δ_{0}Q < q ⩽ Q, p_{1}/q ∈ I , |f(p_{1}/q) − p_{2}/q| < ψ(Q)/Q} .

Proof of Theorem 6 modulo Thorem 7. This is trivial. Given the hypotheses of Theorem 7, the hypotheses of Theorem 6 are clearly satisfied. Fix an interval I ⊆ I_{0}. By Theorem 7, there exist constants δ_{0} and C_{1} so that for all Q sufficiently large

#A_{Q}(I)·\frac{2C_{1}}{Q^{2}ψ(Q)} ≥ ∑_{p/q∈A_{Q}(I)} |B (\frac{p_{1}}{q} , \frac{C_{1}}{Q^{2}ψ(Q)})| ≥ | ⋃_{p/q∈A_{Q}(I)}(B (\frac{p_{1}}{q} , \frac{C_{1}}{Q^{2}ψ(Q)}) ∩ I)| ⩾ \frac{|I|}{2}.

We have that N_{f}(Q, ψ, I) ⩾ #A_{Q}(I) and Theorem 6 follows. ♠

The following corollary of Theorem 7 is crucial for proving Theorems 1 and 4.

Corollary 5 Let ψ and f be as in Theorem 7 and C := {(x, f(x)) : x ∈ I_{0}}. With reference to the ubiquitous framework of §3.2, set

β : J := ℤ^{2} × ℕ → ℕ : (p, q) → q , Φ : t → t^{−1}ψ(t) and ρ : t → u(t)/(t^{2}ψ(t)) (11)

where u : ℝ^{+} → ℝ^{+} is any function such that lim_{t→∞} u(t) = ∞. Then the system (ℚ_{C}^{2}(Φ), β) is locally ubiquitous with respect to ρ.

Remark. Given α = (p, q) ∈ J, the associated resonant point R_{α} in the above ubiquitous system is simply the rational point p/q in the plane. Furthermore, R := ℚ^{2}.

Proof of Corollary 5. For an interval I ⊆ I_{0}, let

A_{Q}^{∗}(I) := {p/q ∈ ℚ^{2} : Q/u(Q) < q ⩽ Q , p_{1}/q ∈ I , |f(p_{1}/q) − p_{2}/q| < ψ(Q)/Q} .

For any δ_{0} ∈ (0, 1), we have that 1/u(Q) < δ_{0} for Q sufficiently large since lim_{t→∞} u(t) = ∞. Thus, for Q sufficiently large, A_{Q}(I) ⊂ A_{Q}^{∗}(I) and Theorem 7 implies that

| ⋃_{p/q∈A_{Q}^{∗}(I)} (B (\frac{p_{1}}{q} , \frac{u(Q)}{Q^{2}ψ(Q)}) ∩ I)| ⩾ | ⋃_{p/q∈A_{Q}(I)} (B (\frac{p_{1}}{q} , \frac{C_{1}}{Q^{2}ψ(Q)}) ∩ I)| ⩾ \frac{|I|}{2}.

This establishes the corollary. ♠

4.2 An auxiluary lemma

The following lemma is an immediate consequence of Theorem 1.4 in [12].

Lemma 5 Let g := (g_{1}, g_{2}) : I_{0} → ℝ^{2} be a C^{(2)} map such that (g′_{1}g′′_{2} − g′_{2}g′′_{1})(x_{0}) ≠ 0 for some point x_{0} ∈ I_{0}. Given positive real numbers δ, K, T and an interval I ⊆ I_{0}, let B(I, δ, K, T) denote the set of x ∈ I for which there exists (q, p_{1}, p_{2}) ∈ ℤ^{3} ∖ {0} satisfying the following system of inequalities:

\{ \begin{matrix}|q g_{1}(x) + p_{1} g_{2}(x) + p_{2}| ⩽ δ \\ |q g′_{1}(x) + p_{1} g′_{2}(x)| ⩽ K \\ |q| ⩽ T . \end{matrix}

Then there is a sufficiently small η = η(x_{0}) > 0 so that for any interval I ⊂ (x_{0} − η, x_{0} + η) there exists a constant C > 0 such that for

0 < δ ⩽ 1, T ⩾ 1, K > 0 and δKT ⩽ 1  (12)

one has

|B(I, δ, K, T)| ⩽ C max (δ^{1/3}, (δKT)^{1/9}) |I|.  (13)

Note that the constant C depends on the interval I. We now show that under the assumption that g is non-degenerate everywhere, the above lemma can be extended to a global statement in which I is any sub-interval of I_{0}.

Lemma 6 Assume that the conditions of Lemma 5 are satisfied and that (g′_{1}g′′_{2} − g′_{2}g′′_{1})(x) ≠ 0 for all x ∈ I_{0}. Then for any finite interval I ⊆ I_{0} there is a constant C > 0 such that for any δ, K, T satisfying (12) one has the estimate (13).

Proof of Lemma 6. As I is a finite interval, its closure \overline{I} is compact. By Lemma 5, for every point x ∈ \overline{I} there is an interval B(x, η(x)) centred at x such that for any sub-interval J of B(x, η(x)) there is a constant C = C_{J} (dependent on J) satisfying (13) with δ, K, T satisfying (12). Since \overline{I} is compact, there is a finite cover \{I_{i} := B(x_{i}, η(x_{i})) : i = 1, …, n\} of \overline{I}. Choose this cover so that n is minimal. Then any interval in this cover is not contained in the union of the others. Otherwise, we would be able to choose another cover with smaller n. We show that any three intervals of this minimal cover do not intersect. Assume the contrary. So there is an x ∈ (a_{1}, b_{2})∩(a_{2}, b_{2})∩(a_{3}, b_{3}), where (a_{i}, b_{i}), i = 1, 2, 3 are intervals of the minimal cover. Then a_{i} < x < b_{i} for each i. Without loss of generality, assume that a_{1} ⩽ a_{2} ⩽ a_{3}. If b_{2} < b_{3} then (a_{2}, b_{2}) ⊂ (a_{1}, b_{3}) = (a_{1}, b_{1}) ∩ (a_{3}, b_{3}), which contradicts the minimality of the cover. Similarly, if b_{3} ⩽ b_{2} then (a_{3}, b_{3}) ⊂ (a_{1}, b_{2}) = (a_{1}, b_{1}) ∩ (a_{2}, b_{2}), a contradiction. This means that the multiplicity of the cover is at most 2. Hence ∑_{i=1}^{n} |I_{i}| ⩽ 2|I|, where I_{i} := B(x_{i}, η(x_{i}). This together with Lemma 5 implies that

|B(I, δ, K, T)| = | ⋃_{i=1}^{n}B(I_{i}, δ, K, T) | ≤ ∑_{i=1}^{n} |B(I_{i}, δ, K, T)|
⩽ ∑_{i=1}^{n}C_{I_{i}} max (δ^{1/3}, (δKT)^{1/9}) |I_{i}|
⩽ max_{i=1,...,n} C_{I_{i}} · max (δ^{1/3}, (δKT)^{1/9}) ∑_{i=1}^{n} |I_{i}|
⩽ 2 max_{i=1,...,n} C_{I_{i}} · max (δ^{1/3}, (δKT)^{1/9}) |I| ,

as required. ♠

4.3 Proof of Theorem 7

Define g(x) := (g_{1}(x), g_{2}(x)) by setting g_{1}(x) := xf′(x) − f(x) and g_{2}(x) := −f′(x). Then g ∈ C^{(2)}. Also, note that

g′(x) = (xf′′(x), −f′′(x)) ,  g′′(x) = (f′′(x) + xf′′′(x), −f′′′(x))  (14)

and

(g′_{1}g′′_{2} − g′_{2}g′′_{1})(x) = f′′(x)^{2} .

As f′′(x) ≠ 0 everywhere, Lemma 6 is applicable to this g. In view of the conditions on the theorem,

sup_{x∈I_{0}} |g′_{2}(x)| = sup_{x∈I_{0}} |f′′(x)| ⩽ c_{2}.  (15)

Define δ_{0} := min{1, (2^{19}c_{2}C^{9})^{−1}}, where C is the constant appearing in Lemma 6. Without loss of generality, assume that C > 1.

Next, fix an interval I ⊆ I_{0}. By Minkowski’s linear forms theorem in the geometry of numbers, for any x ∈ I and Q ∈ ℕ there is a solution (q, p_{1}, p_{2}) ∈ ℤ^{3} ∖ {0} to the system

\{ \begin{matrix}|qg_{1}(x) + p_{1}g_{2}(x) + p_{2}| ⩽ δ_{0}ψ(Q) \\ |qg′_{1}(x) + p_{1}g′_{2}(x)| ⩽ c_{2}(δ_{0}Qψ(Q))^{−1} \\ 0 ⩽ q ⩽ Q . \end{matrix}  (16)

By definition, the set B(I, δ, K, T) with

δ := δ_{0}ψ(Q), K := c_{2}(δ_{0}Qψ(Q))^{−1}, T := 2δ_{0}Q  (17)

consists of points x ∈ I such that there exists a non-zero integer solution (q, p_{1}, p_{2}) to the system (16) with q ⩽ 2δ_{0}Q. By Lemma 6, for sufficiently large Q we have that

|B(I, δ, K, T)| ⩽ C |I| max {(δ_{0}ψ(Q))^{1/3} , (δ_{0}ψ(Q)c_{2}(δ_{0}Qψ(Q))^{−1}2δ_{0}Q)^{1/9}}
= C (2c_{2}δ_{0})^{1/9}|I| ⩽ |I|/4 .

Therefore, with δ, K, T given by (17) and Q sufficiently large

|\frac{3}{4}I \ B(I, δ, K, T)| ⩾ |I|/2 ,  (18)

where \frac{3}{4}I is the interval I scaled by \frac{3}{4}. Notice, that for x ∈ \frac{3}{4}I \ B(I, δ, K, T) we have that

q > 2δ_{0}Q  (19)

for any solution (q, p_{1}, p_{2}) of (16). From now on, assume that x ∈ \frac{3}{4}I \ B(I, δ, K, T). In view of (14) and the second inequality of (16) we have that

|qxf′′(x) − p_{1}f′′(x)| < c_{2}(δ_{0}Qψ(Q))^{−1}.

This together with (19) and the fact that |f′′(x)| > c_{1}, implies that

|x − \frac{p_{1}}{q} | ⩽ \frac{c_{2}}{q|f′′(x)|δ_{0}Qψ(Q)} < \frac{c_{2}}{c_{1}δ_{0}^{2}Q^{2}ψ(Q)} = \frac{C_{1}}{Q^{2}ψ(Q)} ,  (20)

where C_{1} := \frac{c_{2}}{c_{1}δ_{0}^{2}} . In view of (4) and the fact that x ∈ \frac{3}{4}I, we have that p_{1}/q ∈ I for Q is sufficiently large. By Taylor’s formula,

f(\frac{p_{1}}{q} ) = f(x) + f′(x)(\frac{p_{1}}{q} − x) + \frac{1}{2}f′′(x̃)(\frac{p_{1}}{q} − x)^{2}

for some x̃ between x and p_{1}/q. Thus x̃ ∈ I. Now the expression on the left hand side of the first inequality of (16) is equal to

|q(xf′(x) − f(x)) − p_{1}f′(x) + p_{2}| = |(qx − p_{1})f′(x) + p_{2} − qf(x)|
= |(qx − p_{1})f′(x) + p_{2} − q(f(\frac{p_{1}}{q} ) − f′(x)(\frac{p_{1}}{q} − x) − \frac{1}{2}f′′(x̃)(\frac{p_{1}}{q} − x)^{2})|
= |p_{2} − qf(\frac{p_{1}}{q} ) + \frac{q}{2}f′′(x̃)(x − \frac{p_{1}}{q})^{2}| .

It follows from (4), (15), (16) and (20) that for Q sufficiently large

|qf(\frac{p_{1}}{q} ) − p_{2}| ⩽ |p_{2} − qf(\frac{p_{1}}{q} ) + \frac{q}{2}f′′(x̃)(x − \frac{p_{1}}{q})^{2}| + |\frac{q}{2}f′′(x̃)(x − \frac{p_{1}}{q})^{2}|

⩽ δ_{0}ψ(Q) + \frac{Q}{2}c_{2}(\frac{C_{1}}{Q^{2}ψ(Q)})^{2} < 2δ_{0}ψ(Q) .

This inequality together with (19) implies that

|f(\frac{p_{1}}{q} ) − \frac{p_{2}}{q} | < \frac{2δ_{0}ψ(Q)}{q} < \frac{ψ(Q)}{Q} .  (21)

Thus, for any x ∈ \frac{3}{4}I ∖ B(I, δ, K, T) conditions (20) and (21) hold for some (p_{1}, p_{2})/q with 2δ_{0} q < q ≤ Q. Thus, p/q := (p_{1}, p_{2})/q ∈ A_{Q}(I) and moreover, in view of (18) we have that

|_{⋃p/q∈A_{Q}(I)}\{x ∈ I : |x − \frac{p_{1}}{q} | < \frac{C_{1}}{Q^{2}ψ(Q)}\}| ⩾ |I|/2 ,

for all sufficiently large Q. The statement of the theorem now follows. ♠

5. Proof of Theorem 4

Throughout, ψ is an approximating function with λ_{ψ} := lim inf_{t→∞} \frac{− log ψ(t)}{log t} ∈ (1/2, 1). It is readily verified that for any ε > 0

ψ(t) ⩽ t^{−λ_{ψ}+ε}  for all but finitely many t ∈ ℕ ,  (22)

and that there exists a strictly increasing sequence of natural numbers t_{i} such that

ψ(t_{i}) ⩾ t_{i}^{−λ_{ψ}−ε}  for all i ∈ ℕ .  (23)

The dimension part of Theorem 4 is obtained by considering upper and lower bounds separately.

The upper bound. First notice that since f is continuously differentiable the map x ↦ (x, f(x)) is locally bi-Lipshitz and thus preserves Hausdorff dimension [19, 25]. Hence, we will investigate dim Ω_{f,ψ} instead of dim C_{f} ∩ S_{2}(ψ), where Ω_{f,ψ} is defined to be the set of x ∈ I_{0} such that the system of inequalities

\{ \begin{matrix}|x − \frac{p_{1}}{q} | < \frac{ψ(q)}{q}, \\ |f(x) − \frac{p_{2}}{q} | < \frac{ψ(q)}{q} \end{matrix}  (24)

is satisfied for infinitely many p/q ∈ ℚ^{2}. Furthermore, there is no loss of generality in assuming that p_{1}/q ∈ I_{0} for solutions p/q of (24).

Next, without loss of generality, we can assume that I_{0} is open in ℝ. Notice that the set B := {x ∈ I_{0} : |f′′(x)| = 0} is closed in I_{0}. Thus the set G := I_{0} \ B := {x ∈ I_{0} : |f′′(x)| ≠ 0} is open and a standard argument allows one to write G as a countable union of intervals I_{i} on which f satisfies (2) with I_{0} replaced by I_{i}. Of course, the constants c_{1} and c_{2} appearing in (2) will depend on the particular interval I_{i}. The upper bound result will follow on showing that dim Ω_{f,ψ} ∩ I_{i} ≤ d, since by the conditions imposed on the theorem dim B ≤ d and so

dim Ω_{f,ψ} ≤ dim (B ∪ ⋃_{i=1}^{∞} (Ω_{f,ψ} ∩ I_{i})) ≤ d .

Without loss of generality, and for the sake of clarity we assume that f satisfies (2) on I_{0}.

For a point p/q ∈ ℚ^{2}, denote by σ(p/q) the set of x ∈ I_{0} satisfying (24). Trivially, |σ(p/q)| ⩽ 2ψ(q)/q. Assume that σ(p/q) ≠ ∅ and let x ∈ σ(p/q). By the mean value theorem, f(x) = f(p_{1}/q) + f′(x̃)(x − p_{1}/q) for some x̃ ∈ I_{0}. We can assume that f′ is bounded on I_{0} since f′′ is bounded and I_{0} is a bounded interval. Suppose 2^{t} ⩽ q < 2^{t+1}. By (24),

|f(\frac{p_{1}}{q}) − \frac{p_{2}}{q} | ⩽ |f(x) − \frac{p_{2}}{q} | + |f′(x̃)(x − \frac{p_{1}}{q} )| ⩽ c_{4}ψ(q)/q ⩽ c_{4}ψ(2^{t})/2^{t}

where c_{4} > 0 is a constant. In view of (22), this implies that for any ε ∈ (0, 1) and t sufficiently large

|f(\frac{p_{1}}{q}) − \frac{p_{2}}{q} | ⩽ 4c_{4} 2^{(t+1)(−λ_{ψ}+ε)}/2^{t+1} .

By (3), for t sufficiently large the number of p/q ∈ ℚ^{2} with 2^{t} ⩽ q < 2^{t+1} and σ(p/q) ≠ ∅ is at most 2^{t(2−λ_{ψ}+3ε)}. Therefore, with η := (2 − λ_{ψ} + 4ε)/(λ_{ψ} + 1 − ε) it follows that

∑_{p/q∈ℚ^{2} : σ(p/q)≠∅}|σ(p/q)|^{η} = ∑_{t=0}^{∞}  ∑_{p/q∈ℚ^{2}, σ(p/q)≠∅, 2^{t}⩽q<2^{t+1}} |σ(p/q)|^{η}

⩽ c′ ∑_{t=0}^{∞} 2^{t(−λ_{ψ}−1+ε)η} · 2^{t(−λ_{ψ}+2+3ε)} = c′ ∑_{t=0}^{∞} 2^{−tε} < ∞ ,

where c′ is a positive constant. By the Hausdorff–Cantelli Lemma [11, p. 68], dim Ω_{f,ψ} ⩽ η. As ε > 0 is arbitrary,

dim C_{f} ∩ S_{2}(ψ) = dim Ω_{f,λ_{ψ}} ⩽ d := \frac{2 − λ_{ψ}}{λ_{ψ} + 1} .  (25)

The lower bound (modulo Theorem 3). This is a simple consequence of Theorem 3 and so all that is required is that the curve is non-degenerate at a single point. Fix ϵ > 0 such that λ_{ψ} + ϵ < 1 and let

s := \frac{2 − λ_{ψ} − ϵ}{1 + λ_{ψ} + ϵ} < d .

Clearly, s ∈ (1/2, 1). In view of (23) and the fact that ψ is decreasing, there exists a strictly increasing sequence m_{i} of natural numbers such that

ψ(2^{m_{i}}) ≥ 2^{−(λ_{ψ}+ϵ)} 2^{−m_{i}(λ_{ψ}+ϵ)} .  (26)

To see that this is the case, notice that for each t_{i} there exists a natural number m_{i} such that 2^{m_{i}} < t_{i} ≤ 2^{m_{i}+1}. It follows that ψ(2^{m_{i}}) ≥ ψ(t_{i}) ≥ t_{i}^{−(λ_{ψ}+ϵ)} ≥ 2^{−(m_{i}+1)(λ_{ψ}+ϵ)} and to ensure that m_{i−1} < m_{i} simply choose a suitable subsequence. By (26) and the fact that ψ is decreasing, we obtain that

∑_{h=1}^{∞} h^{1−s}ψ(h)^{s+1} = ∑_{t=1}^{∞} ∑_{2^{t−1}⩽h<2^{t}} h^{1−s} ψ(h)^{s+1} ≫ ∑_{t=1}^{∞} 2^{t(2−s)} ψ(2^{t})^{s+1}

≫ ∑_{i=1}^{∞} 2^{m_{i}(2−s)} 2^{−m_{i}(λ_{ψ}+ϵ)(s+1)} = ∞ .

Hence, Theorem 3 implies that H^{s}(C_{f} ∩ S_{2}(ψ)) = ∞ and so dim C_{f} ∩ S_{2}(ψ) ≥ s. As ϵ > 0 can be made arbitrarily small, we obtain the required lower bound result.

The Hausdorff measure part of Theorem 4 is a direct consequence of Theorem 3. Simply note that if lim sup_{h→∞} h^{2−d}ψ(h)^{d+1} > 0 then ∑ h^{1−d}ψ(h)^{d+1} = ∞ and also that if λ_{ψ} ∈ (1/2, 1) then d ∈ (1/2, 1). The latter is obvious. The former follows by first observing that if lim sup_{h→∞} h^{2−d}ψ(h)^{d+1} > 0, then there exists a strictly increasing sequence m_{i} of natural numbers such that 2^{m_{i}(2−d)} ψ(2^{m_{i}})^{d+1} ≥ η > 0. It follows that

∑_{h=1}^{∞} h^{1−d}ψ(h)^{d+1} ≫ ∑_{t=1}^{∞} 2^{t(2−d)} ψ(2^{t})^{d+1} ≫ ∑_{i=1}^{∞} 2^{m_{i}(2−d)}ψ(2^{m_{i}})^{d+1} = ∞ ,

as required. ♠

Alternatively, the lower bound result for dim C_{f} ∩S_{2}(ψ) and the Hausdorff measure part of Theorem 4 can be deduced independently of Theorem 3 via Corollary 4. Note that the upper bound result is complete. It has been established without reference to any other result.

6. Proof of Theorem 1

As C := C_{f} is non-degenerate almost everywhere, we can restrict our attention to a sufficiently small patch of C, which can be written as {(x, f(x)) : x ∈ I} where I is a sub-interval of I_{0} and f satisfies (2) with I_{0} replaced by I. Clearly, Theorem 4 is applicable to f restricted to I. However, without loss of generality and for clarity, we assume that f satisfies (2) on I_{0}.

Throughout this section, ψ will be an approximating function such that

∑_{h=1}^{∞} ψ^{2}(h) = ∞ .  (27)

Step 1. We show that there is no loss of generality in assuming that

ψ(h) ⩽ h^{−1/2}  for all h.  (28)

Define the auxiliary function ψ̃ : h → ψ̃(h) := min{h^{−1/2}, ψ(h)}. Clearly ψ̃ is an approximating function. First we show that

∑_{h=1}^{∞} ψ̃^{2}(h) = ∞ .  (29)

Assume that (29) is false. Then using the fact that ψ̃ is decreasing, we obtain

0 ⟵_{l→∞} ∑_{[l/2]⩽h<l} ψ̃^{2}(h) ⩾ ∑_{[l/2]⩽h<l} ψ̃^{2}(l) ⩾ ψ̃^{2}(l)l/3 .

Thus, ψ̃(l)l^{1/2} → 0 as l → ∞. It follows that ψ̃(l) = o(l^{−1/2}) and so ψ̃(l) = ψ(l) for all but finitely many l. This together with (27) implies (29), a contradiction.

By definition, S_{2}(ψ̃) ⊆ S_{2}(ψ). Thus to complete the proof of Theorem 1 it suffices to prove the result with ψ replaced by ψ̃. Hence, without loss of generality, (28) can be assumed.

Step 2. We show that there is no loss of generality in assuming that

ψ(h) ⩾ h^{−2/3} for all h.  (30)

To this end, define ψ̂ : h → ψ̂(h) := max{ψ(h), h^{−2/3}}. It is readily verified that

S_{2}(ψ̂) = S_{2}(ψ) ∪ S_{2}(h ↦ h^{−2/3}).

By the upper bound result established in §5, we have that dim C_{f} ∩ S_{2}(h ↦ h^{−2/3}) ≤ 4/5 < 1. It follows from the definition of Hausdorff dimension that H^{1}(C_{f} ∩ S_{2}(h ↦ h^{−2/3})) = 0; i.e. for almost all x ∈ I_{0}

(x, f(x)) ∉ S_{2}(h ↦ h^{−2/3}) .

Thus,

|{x ∈ I_{0} : (x, f(x)) ∈ S_{2}(ψ̂)}| = |{x ∈ I_{0} : (x, f(x)) ∈ S_{2}(ψ)}|

and to complete the proof of Theorem 1 it suffices to prove that the set on the left has full measure. Hence, without loss of generality, (30) can be assumed. Step 3. In view of Steps 1 and 2 above, the function ψ satisfies (4) and Corollary 5 is applicable to ψ. By (27) and the fact that ψ is decreasing, we obtain that

∞ = ∑_{t=0}^{∞} ∑_{2^{t}⩽h<2^{t+1}} ψ(h)^{2} ⩽ ∑_{t=0}^{∞} ∑_{2^{t}⩽h<2^{t+1}} ψ(2^{t})^{2} = ∑_{t=0}^{∞} 2^{t}ψ(2^{t})^{2} .

Hence

∑_{t=0}^{∞} 2^{t}ψ(2^{t})^{2} = ∞.

Next, define the increasing function u : ℝ^{+} → ℝ^{+} as follows

u(h) := ∑_{t=0}^{[h]} 2^{t} ψ(2^{t})^{2} .

Trivially, lim_{t→∞} u(t) = ∞. Let a_{t} = 2^{t}ψ(2^{t})^{2} and u_{t} = u(t). Fix k ∈ ℕ. Then

∑_{t=k}^{m} \frac{a_{t}}{u_{t}} ⩾ ∑_{t=k}^{m} \frac{a_{t}}{u_{m}} = \frac{u_{m} − u_{k−1}}{u_{m}} → 1 as m → ∞.

Hence

∑_{t=k}^{∞} \frac{a_{t}}{u_{t}} ⩾ 1 for all k.

This implies that the sum ∑_{t=1}^{∞} a_{t}/u_{t} diverges; i.e.

∑_{t=0}^{∞} \frac{2^{t} ψ(2^{t})^{2}}{u(t)} = ∞ .  (31)

Now let Ψ(t) = Φ(t) := ψ(t)/t and ρ(t) := u(log_{2} t)/(t^{2}ψ(t)). By Corollary 5, (ℚ_{C}^{2}(Φ), β) is locally ubiquitous relative to ρ, where the function β is given by (11). In view of (31),

∑_{t=1}^{∞} \frac{Ψ(2^{t})}{ρ(2^{t})} := ∑_{t=1}^{∞} \frac{\frac{ψ(2^{t})}{2^{t}}}{\frac{u(t)}{2^{2t}ψ(2^{t})}} = ∑_{t=1}^{∞} \frac{2^{t} ψ(2^{t})^{2}}{u(t)} = ∞ .

Since ψ is decreasing,

Ψ(2^{t+1}) := \frac{ψ(2^{t+1})}{2^{t+1}} ⩽ \frac{1}{2} · \frac{ψ(2^{t})}{2^{t}} := \frac{1}{2} Ψ(2^{t}) .

Thus the conditions of Lemma 3 are satisfied and it follows that the set Λ(ℚ_{C}^{2}(Φ), β, Ψ) has full measure. By definition, the set Λ(ℚ_{C}^{2}(Φ), β, Ψ) consists of points x ∈ I_{0} such that the system of inequalities

\{ \begin{matrix}|x − \frac{p_{1}}{q} | < Ψ(q) = \frac{ψ(q)}{q} < \frac{2ψ(q)}{q} \\ |f(x) − \frac{p_{2}}{q} | < Ψ(q) + Φ(q) = \frac{ψ(q)}{q} + \frac{ψ(q)}{q} ⩽ \frac{2ψ(q)}{q} \end{matrix}

is satisfied for infinitely many p/q ∈ ℚ^{2}. Obviously, for x ∈ Λ(ℚ_{C}^{2}(Φ), β, Ψ) the point (x, f(x)) is in S_{2}(2ψ). In order to complete the proof of Theorem 1, simply apply what has already been proved to the approximating function \frac{1}{2}ψ. ♠

7. Proof of Theorem 3

We are assuming that there exists at least one point on the curve C_{f} which is non-degenerate. Thus, there exists a sufficiently small patch of C_{f}, which can be written as {(x, f(x)) : x ∈ I} where I is a sub-interval of I_{0} and f satisfies (2) with I_{0} replaced by I. Clearly, Theorems 1 and 4 are applicable to f restricted to I. However, without loss of generality and for the sake of clarity, we assume that f satisfies (2) on I_{0}.

Throughout this section, s ∈ (1/2, 1) and ψ will be an approximating function such that

∑_{h=1}^{∞} h^{1−s} ψ(h)^{s+1} = ∞ .  (32)

Step 1. We show that there is no loss of generality in assuming that

lim_{t→∞} ψ(t) = 0 .  (33)

Suppose on the contrary that lim sup_{t→∞} ψ(t) > 0. Then for any s ≤ 1, we have that (32) holds. In particular, ∑_{h=1}^{∞} ψ^{2}(h) = ∞ and so Theorem 1 implies that H^{1}(C_{f} ∩ S_{2}(ψ)) > 0. It follows that H^{s}(C_{f} ∩ S_{2}(ψ)) = ∞ for any s < 1. Hence, (33) can be assumed.

Step 2. Since s > 1/2, there exists η > 0 such that s = \frac{1}{2} + η. We show that there is no loss of generality in assuming that for all h ∈ ℕ,

ψ(h) ⩾ h^{−(1−ϵ)} where 0 < ϵ < 4η/(3 + 2η) .  (34)

To this end, define ψ̂ : h → ψ̂(h) := max{ψ(h), h^{−(1−ϵ)}}. It is readily verified that

S_{2}(ψ̂) = S_{2}(ψ) ∪ S_{2}(h ↦ h^{−(1−ϵ)}).

By the upper bound result established in §5, dim C_{f} ∩ S_{2}(h ↦ h^{−(1−ϵ)}) ≤ (1 + ϵ)/(2 − ϵ) < s and so H^{s}(C_{f} ∩ S_{2}(h ↦ h^{−(1−ϵ)})) = 0. Thus,

H^{s}(C_{f} ∩ S_{2}(ψ̂)) = H^{s}(C_{f} ∩ S_{2}(ψ))

and to complete the proof of Theorem 3 it suffices to prove that H^{s}(C_{f} ∩ S_{2}(ψ̂)) = ∞. Hence, without loss of generality, (34) can be assumed. Step 3. In view of Steps 1 and 2 above, the function ψ satisfies (4) and Corollary 5 is applicable to ψ. In view of (32), we can find a strictly increasing sequence of positive integers {h_{i}}_{i∈ℕ} such that

∑_{h_{i−1}<h≤ h_{i}} h^{1−s} ψ(h)^{s+1} > 1 (h_{0} := 0) .

Now simply define the increasing function u as follows:

u : h → u(h) := i for h_{i−1} < h ≤ h_{i} .

Note that

∑_{h=1}^{∞} h^{1−s} ψ(h)^{s+1} u(h)^{−1} = ∑_{i=1}^{∞} ∑_{h_{i−1}<h≤ h_{i}} h^{1−s} ψ(h)^{s+1} u(h)^{−1} > ∑_{i=1}^{∞} i^{−1} = ∞ .

In particular, since the function ψ^{s+1}/u is decreasing we have that

∞ = ∑_{t=0}^{∞} ∑_{2^{t}≤h<2^{t+1}} h^{1−s} ψ(h)^{s+1} u(h)^{−1} ≤ 2^{2−s} ∑_{t=0}^{∞} 2^{t(2−s)} ψ(2^{t})^{1+s} u(2^{t})^{−1} .

Hence

∑_{t=0}^{∞} 2^{t(2−s)} ψ(2^{t})^{1+s} u(2^{t})^{−1} = ∞ .  (35)

Now let Ψ(t) = Φ(t) := ψ(t)/t and ρ(t) := u(t)/(t^{2}ψ(t)). By Corollary 5, (ℚ_{C}^{2}(Φ), β) is locally ubiquitous relative to ρ, where the function β is given by (11). In view of (35),

∑_{t=1}^{∞} \frac{Ψ(2^{t})^{s}}{ρ(2^{t})} := ∑_{t=1}^{∞} 2^{t(2−s)} ψ(2^{t})^{1+s} u(2^{t})^{−1} = ∞ .

Since ψ is decreasing, Ψ(2^{t+1}) ⩽ \frac{1}{2} Ψ(2^{t}). Thus the conditions of Lemma 4 are satisfied and it follows that the set Λ(ℚ_{C}^{2}(Φ), β, Ψ) is of infinite s-dimensional Hausdorff measure. The statement of Theorem 3 now follows on repeating verbatim the argument given towards the end of the proof of Theorem 1. ♠

8. Various Generalizations

8.1 Theorem 3 for general Hausdorff measures

A dimension function h : ℝ^{+} → ℝ^{+} is an increasing, continuous function such that h(r) → 0 as r → 0 . Let H^{h} denote the Hausdorff h–measure with respect to the dimension function h. With reference to §2.2, this measure is defined by replacing l^{s}(C_{i}) in the definition of s– dimensional Hausdorff measure H^{s} by the quantity h(l(C_{i})) – see [19, 25] for further details. In the case that h : r → r^{s} (s ≥ 0), the measure H^{h} is precisely H^{s}. For Hausdorff h–measures, Jarník’s Theorem can be stated as follows – see [6, §1.2 and §12.1].

Jarník’s General Theorem (1931). Let h be a dimension function such that r^{−1} h(r) → ∞ as r → 0 and r^{−1} h(r) is decreasing. Let ψ be an approximating function. Then

H^{h} (S_{n}(ψ)) = \{ \begin{matrix}0 & if ∑_{r=1}^{∞} r^{n} h (ψ(r)/r) < ∞ \\ ∞ & if ∑_{r=1}^{∞} r^{n} h (ψ(r)/r) = ∞ \end{matrix} .

In the most simplest form, the following statement is the Hausdorff h–measure analogue of Theorem 3.

Theorem 8 Let f ∈ C^{(3)}(I_{0}), where I_{0} is an interval and C_{f} := {(x, f(x)) : x ∈ I_{0}}. Assume that there exists at least one point on the curve C_{f} which is non-degenerate. Let ψ be an approximation function and let h be a dimension function such that r^{−1} h(r) → ∞ as r → 0, r^{−1} h(r) is decreasing and r^{−(1/2+ϵ)} h(r) → 0 as r → 0 for ϵ > 0 sufficiently small. Furthermore, suppose h satisfies the following growth condition: there exist constants r_{0}, λ_{1}, λ_{2} ∈ (0, 1) such that h(λ_{1}r) ≤ λ_{2} h(r) for r ∈ (0, r_{0}). Then,

H^{h}((C_{f} ∩ S_{2}(ψ))) = ∞  if  ∑_{r=1}^{∞} r ψ(r) h(ψ(r)/r) = ∞ .

Apart from the growth condition imposed on the dimension function, Theorem 8 is the precise analogue of the divergent part of Jarník’s General Theorem for simultaneous Diophantine approximation on planar curves. The growth condition is not particularly restrictive and can be completely removed from the statement of the theorem in the case that G := lim sup_{r→∞} h(ψ(r)/r) ψ(r) r^{2} > 0. Furthermore, when G = 0, if there exists a constant λ ∈ (0, 1) such that ψ(2r) > λψ(r) for all sufficiently large r then the growth condition on h is again redundant.

Notice that if h : r → r^{s} (s ≥ 0), then the growth condition is trivially satisfied and the above theorem reduces to Theorem 3.

Remark on the proof of Theorem 8. The first step is to obtain the analogue of Lemma 4 for general Hausdorff measures. This is easy, it follows directly from Corollary 3 of [6, §5] in the same way that Lemma 4 is deduced from Lemma 2. The proof of Theorem 8 then follows on modifying the argument used to prove Theorem 3 in §7. Note that Corollary 5, the important local ubiquity statement which gives the ‘optimal’ ubiquitous function ρ, is independent of any dimension function. The following useful fact concerning dimension functions is also required: if f and g are two dimension functions such that f(r)/g(r) → 0 as r → 0, then H^{f}(F) = 0 whenever H^{g}(F) < ∞. We leave the details to the reader.

8.2 The Multiplicative Problems/Theory

Given an approximating function ψ, a point y = (y_{1}, …, y_{n}) ∈ ℝ^{n} is called simultaneously multiplicatively ψ–approximable if there are infinitely many q ∈ ℕ such that

∏_{1⩽i⩽n} ∥qy_{i}∥ < ψ(q)^{n}.

Thus, the maximum in the definition of simultaneously ψ–approximable is replaced by the product. Denote by S_{n}^{M}(ψ) the set of simultaneously multiplicatively ψ–approximable points. Trivially, we have that

S_{n}(ψ) ⊂ S_{n}^{M}(ψ) .

The two fundamental problems posed in the introduction can obviously be reinstated for the multiplicative setup. In a forthcoming paper [7], the first and third authors develop the simultaneous multiplicative theory for metric Diophantine approximation on planar curves. As an illustration of the type of results established in [7], we mention the following analogue of Theorem 4. With the same notation and hypotheses of Theorem 4,

dim C_{f} ∩ S_{2}^{M}(ψ) = \frac{2 − λ_{ψ}}{λ_{ψ} + 1} .

Appendix I: Proof of Ubiquity Lemmas

A. Ubiquity with respect to sequences

In this appendix we prove the ubiquity lemmas of §3.1 which are the key towards establishing the divergent results of this paper. It is both more convenient and no more difficult to consider a slightly more general setup in which the sequence {2^{n}} is replaced by an arbitrary increasing sequence u. Apart from this the setup remains unchanged.

Let u := {u_{n}} be a positive increasing sequence such that lim_{n→∞} u_{n} = ∞ and let J^{u}(n) := {α ∈ J : β_{α} ⩽ u_{n}}. Given a function ρ : ℝ^{+} → ℝ^{+} such that lim_{t→∞} ρ(t) = 0, let

Δ^{u}(ρ, n) := ⋃_{α∈J^{u}(n)} B(R_{α}, ρ(u_{n})) .

Definition 4 Suppose there exists a function ρ, a sequence u and an absolute constant κ > 0 such that for any interval I ⊆ I_{0}

|Δ^{u}(ρ, n) ∩ I| ⩾ κ |I|  for n ≥ n_{o}(I).  (36)

Then the pair (R, β) is said to be locally ubiquitous in I_{0} relative to (ρ, u).

Notice that any subsequence v of u will also do in the above definition; i.e. (36) is satisfied for Δ^{v}(ρ, n). In order to state the consequences of this slightly more general definition of ubiquity we introduce the following notion. Given a sequence u, a function h will be said to be u-regular if there exists a strictly positive constant λ < 1 such that for n sufficiently large

h(u_{n+1}) ≤ λ h(u_{n}) .  (37)

The constant λ is independent of n but may depend on u. Clearly, if h is u-regular then it is v–regular for any subsequence v of u.

Theorem 9 Suppose that (R, β) is locally ubiquitous in I_{0} relative to (ρ, u) and let Ψ be an approximating function such that Ψ is u-regular. Then

|Λ(R, β, Ψ)| = FULL := |I_{0}|  if  ∑_{n=1}^{∞} \frac{Ψ(u_{n})}{ρ(u_{n})} = ∞ .

Theorem 10 Suppose that (R, β) is locally ubiquitous in I_{0} relative to (ρ, u) and let Ψ be an approximation function. Let s ∈ (0, 1) and let

G := lim sup_{t→∞} \frac{Ψ(u_{n})^{s}}{ρ(u_{n})} .  (38)

(i) Suppose that G = 0 and that Ψ is u-regular. Then,

H^{s}(Λ(R, β, Ψ)) = ∞  if  ∑_{n=1}^{∞} \frac{Ψ(u_{n})^{s}}{ρ(u_{n})} = ∞ .

(ii) Suppose that G > 0. Then, H^{s}(Λ(R, β, Ψ)) = ∞.

In the case u = {u_{n}} := {2^{n}}, these theorems clearly reduce to Lemmas 1 and 2 of §3.1.

A.1. Prerequisites

A.1.1. The Mass Distribution Principle and a covering lemma

A general and classical method for obtaining a lower bound for the s–dimensional Hausdorff measure of an arbitrary set F is the following mass distribution principle.

Lemma 7 (Mass Distribution Principle) Let μ be a probability measure supported on a subset F of ℝ. Suppose there are positive constants c and r_{o} such that μ(B) ⩽ c r^{s} for any ball B with radius r ≤ r_{o}. Then, H^{s}(F) ≥ 1/c.

The following covering result will be used at various stages during the proof of our theorems.

Lemma 8 (Covering lemma) Let B be a finite collection of balls in ℝ with common radius r > 0. Then there exists a disjoint sub-collection {B_{i}} such that

⋃_{B∈B} B ⊂ ⋃_{i} 3B_{i} .

These lemmas are easily established and relatively standard, see [19, 25] and [6, §7].

A.1.2. Positive and full measure sets

Proposition 1 Let E ⊂ ℝ be a measurable set and let I_{0} ⊂ ℝ be an interval. Assume that there is a constant c > 0 such that for any finite interval I ⊂ I_{0} we have that |E ∩ I| ⩾ c |I|. Then E has full measure in I_{0}, i.e. |I_{0} \ E| = 0.

For the proof see [3, Lemma 2] and [6, §8].

Proposition 2 Let E_{n} ∈ ℝ be a sequence of measurable sets such that ∪_{n=1}^{∞}E_{n} is bounded and ∑_{n=1}^{∞} |E_{n}| = ∞. Then

| lim sup_{n→∞} E_{n}| ≥ lim sup_{Q→∞} \frac{(∑_{s=1}^{Q} |E_{s}|)^{2}}{∑_{s,t=1}^{Q} |E_{s} ∩ E_{t}|} .

This result is a generalization of the divergent part of the standard Borel–Cantelli lemma. For the proof see Lemma 5 in [29].

Proposition 3 Let E_{n} ∈ ℝ be a sequence of measurable sets and let I ⊂ ℝ be a bounded interval. Suppose there exists a constant c > 0 such that lim sup_{n→∞} |I ∩ E_{n}| ⩾ c |I|. Then, |I ∩ lim sup_{n→∞} E_{n}| ⩾ c^{2} |I|.

Proof of Proposition 3. For any 0 < ε < c, there is a subsequence E_{n_{i}} with n_{i} strictly increasing such that |I∩E_{n_{i}}) ⩾ (c−ε)|I|. Clearly ( ∑_{i=1}^{N} |I∩E_{n_{i}}| )^{2} ⩾ ( ∑_{i=1}^{N}(c−ε) |I| )^{2} = (c− ε)^{2}N^{2} |I|^{2} and ∑_{n,m=1}^{N} |I ∩ E_{n} ∩ E_{m}| ⩽ ∑_{m,n=1}^{N} |I| = |I| N^{2}. Also notice that ∑_{i=1}^{∞} |I ∩E_{n_{i}}| ≥ |I| ∑_{i=1}^{∞}(c−ε) = ∞. Thus on applying Proposition 2 and observing that I ∩ lim sup_{n→∞} E_{n} ⊇ I ∩ lim sup_{i→∞} E_{n_{i}} we have that

|I ∩ lim sup_{n→∞} E_{n}| ⩾ lim sup_{N→∞} \frac{(c − ε)^{2}N^{2}|I|^{2}}{|I|N^{2}} = (c − ε)^{2} |I| .

As ε > 0 is arbitrary, this completes the proof of the proposition. ♠

B. Proof of Theorem 9

Let B be an arbitrary ball in I_{0} and let r(B) denote its radius. In view of Proposition 1, the aim is to show that

|Λ(R, β, Ψ) ∩ B| ⩾ |B|/C ,  (39)

where C > 0 is a constant independent of B.

B.1. The subset A(Ψ, B) of Λ(R, β, Ψ) ∩ B

Consider the collection of balls {B(R_{α}, 2ρ(u_{n})) : α ∈ J^{u}(n)}. By the covering lemma, there exists a disjoint sub-collection {B(R_{α}, 2ρ(u_{n})) : α ∈ G^{u}(n)}, where G^{u}(n) is a subset of J^{u}(n), such that

⋃_{α ∈G^{u}(n)}^{∘} B(R_{α}, ρ(u_{n})) ⊂ Δ^{u}(ρ, n) ⊂ ⋃_{α ∈G^{u}(n)} B(R_{α}, 6ρ(u_{n})) .  (40)

The left hand side follows from the fact that the balls B(R_{α}, 2ρ(u_{n})) with α ∈ G^{u}(n) are disjoint. Choose n sufficiently large so that 24ρ(u_{n}) < r(B) (by definition, ρ(u_{n}) → 0 as n → ∞) and let

Gu_{B}(n) := \{α ∈ G^{u}(n) : R_{α} ∈ \frac{1}{2} B\} .

In view of (40),

⋃_{α ∈G_{B}^{u}(n)}^{∘} B(R_{α}, ρ(u_{n})) ⊂ Δ^{u}(ρ, n) ∩ B  (41)

and

⋃_{α ∈G_{B}^{u}(n)} B(R_{α}, 6ρ(u_{n}))) ⊃ Δ^{u}(ρ, n) ∩ \frac{1}{4} B .

We now estimate the cardinality of Gu_{B}(n). By (36), for n sufficiently large

#Gu_{B}(n) ρ(u_{n}) ≫ |⋃_{α∈G_{B}^{u}(n)} B(R_{α}, 6ρ(u_{n}))| ≥ |Δ^{u}(ρ, n) ∩ \frac{1}{4} B| ≫ κ |B| .

On the other hand, |B| ⩾ |⋃^{∘}_{α∈G_{B}^{u}(n)}B(R_{α}, ρ(u_{n}))| ≫ #Gu_{B}(n) ρ(u_{n}). The upshot is that

#Gu_{B}(n) ≍ \frac{|B|}{ρ(u_{n})} .  (42)

Suppose for the moment that for some sufficiently large n ∈ ℕ we have that Ψ(u_{n}) ≥ ρ(u_{n}). Then (36) implies that |Δ^{u}(Ψ, n) ∩ B| ≥ |Δ^{u}(ρ, n) ∩ B| ≥ κ |B|. Thus, if Ψ(u_{n}) ≥ ρ(u_{n}) for infinitely many n ∈ ℕ, Proposition 3 implies (39) and we are done. Hence, without loss of generality, we can assume that for n sufficiently large

ρ(u_{n}) > Ψ(u_{n}) .  (43)

Now let

A_{n}(Ψ, B) := ⋃_{α∈G_{B}^{u}(n)}^{∘} B(R_{α}, Ψ(u_{n})) .

The disjointness is a consequence (43). Indeed, for α ∈ Gu_{B}(n) the balls B(R_{α}, 2ρ(u_{n})) are disjoint. Therefore, |A_{n}(Ψ, B)| ≍ Ψ(u_{n}) #Gu_{B}(n) and in view of (42) we have that

|A_{n}(Ψ, B)| ≍ |B| × \frac{Ψ(u_{n})}{ρ(u_{n})} .  (44)

Finally, let

A(Ψ, B) := lim sup_{n→∞} A_{n}(Ψ, B) := ⋂_{m=1}^{∞} ⋃_{n=m}^{∞} A_{n}(Ψ, B) .

By construction, we have A_{n}(Ψ, B) ⊂ Δ^{u}(Ψ, n) ∩ B and it follows that A(Ψ, B) \ R is a subset of Λ(R, β, Ψ) ∩ B. Now in view of (39) and the fact that R is countable and therefore of measure zero, the proof of Theorem 9 will be completed on showing that

|A(Ψ, B)| = |A(Ψ, B) ∩ B| ⩾ m(B)/C.  (45)

Notice that (44) together with the divergent sum hypothesis of the theorem implies that

∑_{n=1}^{∞} |A_{n}(Ψ, B)| = ∞ .  (46)

In view of Proposition 2, this together with the following quasi-independence on average result implies (45) and thereby completes the proof of Theorem 9.

Lemma 9 (Quasi–independence on average) There exists a constant C > 1 such that for Q sufficiently large,

∑_{s,t=1}^{Q} |A_{s}(Ψ, B) ∩ A_{t}(Ψ, B)| ≤ \frac{C}{|B|} (∑_{s=1}^{Q} |A_{s}(Ψ, B)|)^{2} .

Proof of Lemma 9. Throughout, write A_{t}(Ψ) for A_{t}(Ψ, B). Also, let s < t and note that

|A_{s}(Ψ) ∩ A_{t}(Ψ)| = ∑_{α∈G_{B}^{u}(s)} | B(R_{α}, Ψ(u_{s})) ∩ A_{t}(Ψ) | .  (47)

Let B_{s}(Ψ) denote a generic ball B(R_{α}, Ψ(u_{s})) with α ∈ Gu_{B}(s). We now obtain an upper bound for |B_{s}(Ψ) ∩ A_{t}(Ψ)|. Trivially,

|B_{s}(Ψ) ∩ A_{t}(Ψ)| := |B_{s}(Ψ) ∩ ⋃^{∘}_{α∈G_{B}^{u}(t)} B(R_{α}, Ψ(u_{t}))|
= ∑_{α∈G_{B}^{u}(t)} |B_{s}(Ψ) ∩ B(R_{α}, Ψ(u_{t}))| ≪ N(t, s) Ψ(u_{t})  (48)

where N(t, s) := #{α ∈ Gu_{B}(t) : B_{s}(Ψ) ∩ B(R_{α}, Ψ(u_{t})) ≠ ∅}. We proceed by considering two cases.

Case (i): t > s such that Ψ(u_{s}) < ρ(u_{t}). On using the fact that the balls B(R_{α}, 2ρ(u_{t})) with α ∈ Gu_{B}(t) are disjoint, it is easily verified that N(t, s) ≤ 1. This together with (42), (47) and (48) implies that

|A_{s}(Ψ) ∩ A_{t}(Ψ)| ≪ #Gu_{B}(s) Ψ(u_{t}) ≪ |B| × \frac{Ψ(u_{t})}{ρ(u_{s})} .

Case (ii): t > s such that Ψ(u_{s}) ⩾ ρ(u_{t}). First note that if B_{s}(Ψ) ∩ B(R_{α}, ρ(u_{t})) ≠ ∅, then B(R_{α}, Ψ(u_{t})) ⊂ 3B_{s}(Ψ). The balls B(R_{α}, ρ(u_{t})) with α ∈ Gu_{B}(t) are disjoint and so N(t, s) ≪ Ψ(u_{s})/ρ(u_{t}). It now follows, via (42), (44), (47) and (48), that

|A_{s}(Ψ) ∩ A_{t}(Ψ)| ≪ \frac{1}{|B|} |A_{s}(Ψ)| |A_{t}(Ψ)| .

The upshot of these two cases, is that for Q sufficiently large

∑_{s,t=1}^{Q} |A_{s}(Ψ) ∩ A_{t}(Ψ)| = ∑_{s=1}^{Q} |A_{s}(Ψ)| + 2 ∑_{s=1}^{Q−1} ∑_{\begin{matrix}s+1≤t≤Q \\ case(i) \end{matrix}} |A_{s}(Ψ) ∩ A_{t}(Ψ)|

+ 2 ∑_{s=1}^{Q−1} ∑_{\begin{matrix}s+1≤t≤Q \\ case(ii) \end{matrix}} |A_{s}(Ψ) ∩ A_{t}(Ψ)|

≪ ∑_{s=1}^{Q} |A_{s}(Ψ)| + \frac{1}{|B|}(∑_{s=1}^{Q}|A_{s}(Ψ)|)^{2}

+ |B| ∑_{s=1}^{Q−1} ∑_{\begin{matrix}s+1≤t≤Q \\ Ψ(u_{s})<ρ(u_{t}) \end{matrix}} \frac{Ψ(u_{t})}{ρ(u_{s})} .

We now make use of the fact that Ψ is u-regular. For t > s with s sufficiently large, we have that Ψ(u_{t}) ≤ λ^{t−s}Ψ(u_{s}) for some 0 < λ < 1. This together with (44), implies that for Q sufficiently large

|B| ∑_{s=1}^{Q−1} ∑_{\begin{matrix}s+1≤t≤Q \\ Ψ(u_{s})<ρ(u_{t}) \end{matrix}} \frac{Ψ(u_{t})}{ρ(u_{s})} ≪ |B| ∑_{s=1}^{Q−1} \frac{Ψ(u_{s})}{ρ(u_{s})} ∑_{s < t≤ Q} λ^{t−s} ≪ ∑_{s=1}^{Q} |A_{s}(Ψ)| .

By (46), for Q sufficiently large ∑_{s=1}^{Q} |A_{s}(Ψ)| ⩽ |B|^{−1}(∑_{s=1}^{Q} |A_{s}(Ψ)|)^{2}. The statement of Lemma 9 now readily follows. This completes the proof of Theorem 1. ♠

C. Proof of Theorem 10

To prove Theorem 10 we proceed as follows. For any fixed η ≫ 1 we construct a Cantor subset K_{η} of Λ(R, β, Ψ) and a probability measure μ supported on K_{η} satisfying the condition that for an arbitrary ball A of sufficiently small radius r(A)

μ(A) ≪ \frac{r(A)^{s}}{η} ,  (49)

where the implied constant is absolute. By the Mass Distribution Principle, the above inequality implies that H^{s}(K_{η}) ≫ η. Since K_{η} ⊂ Λ(R, β, Ψ), we obtain that H^{s} (Λ(R, β, Ψ)) ≫ η. However, η ≫ 1 is arbitrarily large whence H^{s} (Λ(R, β, Ψ)) = ∞ and this proves Theorem 10.

In view of the above outline, the whole strategy of our proof is centred around the construction of a ‘right type’ of Cantor set K_{η} which supports a measure μ with the desired property. The actual nature of the construction of K_{η} depends heavily on whether G defined by (38) is finite or infinite. We first deal with the case that 0 ≤ G < ∞. The case that G = ∞ is substantially easier.

C.1. Preliminaries

In this section we group together for clarity and convenience various concepts and results which will be required in constructing the Cantor set K_{η}. Throughout, g will denote the function given by

g(r) := Ψ(r)^{s}ρ(r)^{−1} and so G := lim sup_{n→∞} g(u_{n}) .

C.1.1. The sets Gu_{B}(n) again.

Let B be an arbitrary ball in I_{0} with radius r(B). Relabel the sets Gu_{B}(n) constructed in §B by G̃u_{B}(n). By keeping track of constants, the estimate (42) for #G̃u_{B}(n) is explicitly as follows:

\frac{κ}{24} \frac{r(B)}{ρ(u_{n})} ≤ #G̃u_{B}(n) ≤ \frac{r(B)}{ρ(u_{n})} ,

where κ is as in (36). Now let 0 < c_{1} := \frac{κ}{24} < 1 and define Gu_{B}(n) to be any sub-collection of G̃u_{B}(n) such that

#Gu_{B}(n) = [c_{1} \frac{r(B)}{ρ(u_{n})}] ,

where [x] denotes the integer part of a real number x. Thus, for n sufficiently large

\frac{1}{2} c_{1} \frac{r(B)}{ρ(u_{n})} ≤ #Gu_{B}(n) ≤ c_{1} \frac{r(B)}{ρ(u_{n})} .  (50)

Remark. Recall, that by construction the balls B(R_{α}, 2ρ(u_{n})) with α ∈ Gu_{B}(n) are disjoint. Also note, that we can assume that ρ(u_{n})^{−1} Ψ(u_{n}) → 0 as n → ∞. If this was not the case then lim sup ρ(u_{n})^{−1} Ψ(u_{n}) > 0 as n → ∞ and Theorem 1 implies that |Λ(R, β, Ψ)| = |I_{0}| > 0. In turn, H^{s}(Λ(R, β, Ψ)) = ∞ for any s ∈ [0, 1) and we are done. Hence, without loss of generality, we can assume that for n sufficiently large

2 Ψ(u_{n}) < ρ(u_{n})  and  lim_{n→∞} Ψ(u_{n}) = 0 .  (51)

C.1.2. Working on a subsequence of u and the ubiquity function ρ.

The proof of Theorem 10 in the case that G is finite relies on the fact that the ubiquity function ρ can be taken to be u-regular with constant λ as small as we please. The fact that we have assumed that the approximating function Ψ is u-regular in the hypothesis of the theorem is purely for convenience with the application to planar curves in mind. To begin with recall the following simple facts: (i) if we have local ubiquity for a particular sequence u then we automatically have local ubiquity for any subsequence v and (ii) if a function h is u-regular then it is v-regular for any subsequence v. Also note that if G is finite, then lim sup_{n→∞} g(v_{n}) < ∞ for any subsequence v of u.

Suppose G is finite and fix some λ ∈ (0, 1). We now prove the existence of an appropriate subsequence v of u on which ρ is v-regular with constant λ and ∑ g(v_{n}) = ∞. In the case G = 0 (part (i) of Theorem 10), we have that Ψ is u-regular and so there exists a constant λ_{∗} ∈ (0, 1) such that Ψ(u_{n+1}) ≤ λ_{∗} Ψ(u_{n}) for all n sufficiently large. It follows that for n sufficiently large

x_{n+1} := Ψ(u_{n+1})^{s} ≤ λ_{∗}^{s}Ψ(u_{n})^{s} = λ_{∗}^{s} x_{n} .

Next, fix some sufficiently large n_{1} and for k ≥ 2 let n_{k} be the least integer strictly greater than n_{k−1} such that ρ(u_{n_{k}}) ≤ λ ρ(u_{n_{k−1}}). This is possible since ρ(r) → 0 as r → ∞. By construction, ρ(u_{m}) ≥ λ ρ(u_{n_{k−1}}) for any integer m ∈ [n_{k−1}, n_{k} − 1]. It follows that

∞ = ∑_{n=n_{1}}^{∞} g(u_{n}) = ∑_{n=n_{1}}^{∞} x_{n} ρ(u_{n})^{−1} = ∑_{k=2}^{∞} ∑_{n_{k−1}≤m<n_{k}}x_{m} ρ(u_{m})^{−1}

≤ ∑_{k=2}^{∞} ∑_{n_{k−1}≤m<n_{k}}x_{m} ρ(u_{n_{k−1}})^{−1} λ^{−1} = λ^{−1} ∑_{k=2}^{∞} ρ(u_{n_{k−1}})^{−1} ∑_{n_{k−1}≤m<n_{k}} x_{m}

≪ ∑_{k=2}^{∞} ρ(u_{n_{k−1}})^{−1}x_{n_{k−1}} ∑_{i=0}^{∞}(λ_{∗}^{s})^{i} ≪ ∑_{k=1}^{∞} ρ(u_{n_{k}})^{−1}x_{n_{k}} := ∑_{k=1}^{∞} g(u_{n_{k}}).

Now set v := {u_{n_{k}}}. By construction, ρ is v-regular with constant λ and ∑ g(v_{n}) = ∞. Next notice that if 0 < G < ∞, then there exists a strictly increasing sequence {n_{i}} such that g(u_{n_{i}}) ≥ G/2 > 0. Since lim_{r→∞} ρ(r) = 0, it follows that for any λ < 1 there exists a subsequence v of {u_{n_{i}}} such that ρ(v_{t+1}) < λ ρ(v_{t}) and ∑ g(v_{t}) = ∞. The upshot is that in establishing Theorem 10 for the case that 0 ≤ G < ∞ we can assume that ρ is u regular with constant λ as small as we please.

C.2. Proof of Theorem 10: 0 ≤ G < ∞

C.2.1. The Cantor set K_{η}

Let G^{∗} := max{2, sup_{n∈ℕ} g(u_{n})} and fix a real number η > G^{∗}. Thus

g(u_{n}) < G^{∗} < η for all n .

To avoid cumbersome expressions, let ϖ denote the following repeatedly occurring constant

ϖ := \frac{c_{1}}{96} < 1 .  (52)

In view of the discussion of §C.1.2, we can assume that for n sufficiently large

ρ(u_{n+1}) ≤ λ ρ(u_{n})  with 0 < λ < \frac{1}{9} .  (53)

Constructing the first level K(1).

Choose t_{1} large enough so that

g(u_{t_{1}}) < G^{∗} < \frac{η}{24ϖ} ,  (54)
ψ(u_{t_{1}})^{s−1} > \frac{η}{ϖ} ,  (55)

and so that the counting estimate (50) is valid for the set G_{I_{0}}^{u}(t_{1}); i.e. with B = I_{0}. Note that the first of these inequalities is possible since g(u_{n}) < G^{∗} < η. The latter inequality is possible in view of (51) and since s < 1. Let k_{1} ≥ 1 be the unique integer such that

\frac{6ϖ}{η} ∑_{i=0}^{k_{1}−1} g(u_{t_{1}+i}) ⩽ \frac{1}{4} < \frac{6ϖ}{η} ∑_{i=0}^{k_{1}} g(u_{t_{1}+i}) .  (56)

Note, the fact that k_{1} ≥ 1 is a consequence of (54). The first level K(1) of the Cantor set K_{η} will consist of sub-levels K(t_{1} + i) where 0 ≤ i ≤ k_{1}.

• The sub-level K(t_{1}) : This consists of balls of common radius ψ(u_{t_{1}}) defined as follows:-

K(t_{1}) := ⋃_{α∈V_{I_{0}}^{u}(t_{1})}^{∘} B(R_{α}, ψ(u_{t_{1}})) where V_{I_{0}}^{u}(t_{1}) := G_{I_{0}}^{u}(t_{1}) .

• The sub-levels K(t_{1} + i) for 1 ≤ i ≤ k_{1} : These are constructed inductively. The key to the whole procedure is the existence of ‘special’ subsets V_{I_{0}}^{u}(t_{1} + i) of G_{I_{0}}^{u}(t_{1} + i). Suppose for 0 ≤ j ≤ i − 1 we have constructed the sub-levels

K(t_{1} + j) = ⋃_{α∈V_{I_{0}}^{u}(t_{1}+j)}^{∘} B(R_{α}, ψ(u_{t_{1}+j})) .

We proceed to construct K(t_{1} + i) – equivalently V_{I_{0}}^{u}(t_{1} + i). Let

h(u_{t_{1}+j}) := \frac{ϖ}{η} Ψ(u_{t_{1}+j})^{s} .

Note that in view of (54) and (55) we have that ψ(u_{t_{1}+j}) < h(u_{t_{1}+j}) < ρ(u_{t_{1}+j}). Define

T (t_{1} + j) := {B(R_{α}, h(u_{t_{1}+j})) : α ∈ V_{I_{0}}^{u}(t_{1} + j)} .

Now for each α ∈ G_{I_{0}}^{u}(t_{1} + i) construct the ball B(R_{α}, ρ(u_{t_{1}+i})). Clearly, the balls in this collection are also disjoint and we proceed by disregarding any of those which lie too close to balls from any of the previous sub-levels K(t_{1} + j). To make this precise, we introduce the sets

U_{I_{0}}^{u}(t_{1} + i) := {α ∈ G_{I_{0}}^{u}(t_{1} + i) : B(R_{α}, ρ(u_{t_{1}+i})) ∩ ⋃_{j=0}^{i−1} T (t_{1} + j) ≠ ∅}
V_{I_{0}}^{u}(t_{1} + i) := G_{I_{0}}^{u}(t_{1} + i) \ U_{I_{0}}^{u}(t_{1} + i) .

By construction, V_{I_{0}}^{u}(t_{1} + j) ⊆ G_{I_{0}}^{u}(t_{1} + j) for 0 ⩽ j ⩽ i. In particular, the balls in T (t_{1} + j) are disjoint. Thus, #T (t_{1} + j) = #V_{I_{0}}^{u}(t_{1} + j) ≤ #G_{I_{0}}^{u}(t_{1} + j). We claim that #V_{I_{0}}^{u}(t_{1} + i) ≥ \frac{1}{2} G_{I_{0}}^{u}(t_{1} + i) . There are two cases to consider.

\underline{Case} (i): 0 ≤ j ≤ i − 1 such that ρ(u_{t_{1}+i}) < h(u_{t_{1}+j}). The number of disjoint balls of radius ρ(u_{t_{1}+i}) that can possibly intersect a ball in T (t_{1} + j) is ⩽ 3h(u_{t_{1}+j})/ρ(u_{t_{1}+i}).

\underline{Case} (ii): 0 ≤ j ≤ i − 1 such that ρ(u_{t_{1}+i}) ≥ h(t_{1} + j). In this case, each ball in T (t_{1} + j) can intersect at most one ball B(R_{α}, ρ(u_{t_{1}+i})) with α ∈ G_{I_{0}}^{u}(t_{1} + i). This makes use of the fact that the corresponding enlarged balls B(R_{α}, 2ρ(u_{t_{1}+i})) are disjoint.

It follows that

#U_{I_{0}}^{u}(t_{1} + i) ⩽ ∑_{case (i)} \frac{3ϖ}{η} \frac{ψ(u_{t_{1}+j})^{s}}{ρ(u_{t_{1}+i})} #T (t_{1} + j) + ∑_{case (ii)} #T (t_{1} + j).

Recall that #T (t_{1} + j) ≤ #G_{I_{0}}^{u}(t_{1} + j). Thus, the contribution from the sum over case (i) is:

≤ ∑_{j=0}^{k_{1}−1} \frac{6ϖ}{η} g(u_{t_{1}+j}) #G_{I_{0}}^{u}(t_{1} + i) ⩽ \frac{1}{4} #G_{I_{0}}^{u}(t_{1} + i) ,

by (50) and the choice of k_{1} – see (56). The contribution from the sum over case (ii) is:

≤ ∑_{case (ii)} #G_{I_{0}}^{u}(t_{1} + j) ≤ 2 ∑_{j=0}^{i−1} \frac{ρ(u_{t_{1}+i})}{ρ(u_{t_{1}+j})}#G_{I_{0}}^{u}(t_{1} + i)

≤ 2 #G_{I_{0}}^{u}(t_{1} + i) ∑_{j=0}^{i−1} λ^{i−j} < 2 #G_{I_{0}}^{u}(t_{1} + i) ∑_{s=1}^{∞} λ^{s} < \frac{1}{4} #G_{I_{0}}^{u}(t_{1} + i) ,

by (50) and the choice of λ – see (53). Hence, #U_{I_{0}}^{u}(t_{1} + i) < \frac{1}{2} #G_{I_{0}}^{u}(t_{1} + i) so that

#V_{I_{0}}^{u}(t_{1} + i) ≥ \frac{1}{2} #G_{I_{0}}^{u}(t_{1} + i).  (57)

The sub-level K(t_{1} + i) is defined to be:

K(t_{1} + i) := ⋃_{R_{α}∈V_{I_{0}}^{u}(t_{1}+i)} B(R_{α}, ψ(u_{t_{1}+i})) .

Also, note that by construction K(t_{1} + i) ∩ K(t_{1} + j) = ∅ for 0 ≤ i ≠ j ≤ k_{1}. The first level K(1) of the Cantor set is defined to be

K(1) := ⋃_{i=0}^{k_{1}} K(t_{1} + i) .

Higher levels K(n) and the Cantor set K_{η}.

For any integer n ≥ 2, the n’th level K(n) will be defined recursively in terms of local levels K(n, B) associated with balls B from the previous level K(n − 1):

K(n) := ⋃_{B∈K(n−1)}^{∘} K(n, B) ,

where

K(n, B) := ⋃_{i=0}^{k_{n}(B)} K(t_{n} + i, B) .

To start with, choose t_{n} > t_{n−1} sufficiently large so that for any ball B ∈ K(n−1) the counting estimate (50) is valid and so that

ψ(u_{t_{n}})^{s−1} > \frac{r(B)^{s−1}}{ϖ} .  (58)

In view of (51), (55), the fact that g(u_{n}) < G^{∗} for all n and that s < 1, we have that

g(u_{t_{n}}) < G^{∗} < \frac{r(B)^{s−1}}{24ϖ}  ∀ B ∈ K(n − 1).  (59)

Fix a ball B in K(n − 1) and let k_{n}(B) ≥ 1 be the unique integer such that

\frac{6ϖ}{r(B)^{s−1}} ∑_{i=0}^{k_{n}(B)−1} g(u_{t_{n}+i}) ⩽ \frac{1}{4} < \frac{6ϖ}{r(B)^{s−1}} ∑_{i=0}^{k_{n}(B)} g(u_{t_{n}+i}) .  (60)

The fact that k_{n}(B) ≥ 1 is a consequence of (59). We now construct the local level K(n, B).

• The local sub-level K(t_{n}, B) : Let

K(t_{n}, B) := ⋃_{α∈V_{B}^{u}(t_{n})} B(R_{α}, ψ(u_{t_{n}})) where V_{B}^{u}(t_{n}) := G_{B}^{u}(t_{n}) .

By construction, K(t_{n}, B) ⊂ B – see (41).

• The local sub-levels K(t_{n} + i, B) for 1 ≤ i ≤ k_{n}(B) : Suppose for 0 ≤ j ≤ i − 1 we have constructed the local sub-levels

K(t_{n} + j, B) = ⋃_{α∈V_{B}^{u}(t_{n}+j)}^{∘} B(R_{α}, ψ(u_{t_{n}+j})) .

Let

h_{B}(u_{t_{n}+j}) := \frac{ϖ ψ(u_{t_{n}+j})^{s}}{r(B)^{s−1}} .

In view of (58) and (59) we have that

ψ(u_{t_{n}+j}) < h_{B}(u_{t_{n}+j}) < ρ(u_{t_{n}+j}).  (61)

Define

T (t_{n} + j, B) := {B(R_{α}, h_{B}(u_{t_{n}+j})) : α ∈ V_{B}^{u}(t_{n} + j)} .

Next, introduce the sets

U_{B}^{u}(t_{n} + i) := {α ∈ G_{B}^{u}(t_{n} + i) : B(R_{α}, ρ(u_{t_{n}+i})) ∩ ⋃_{j=0}^{i−1} T (t_{n} + j, B) ≠ ∅}
V_{B}^{u}(t_{n} + i) := G_{B}^{u}(t_{n} + i) \ U_{B}^{u}(t_{n} + i) .

By construction, V_{B}^{u}(t_{n} + j) ⊆ G_{B}^{u}(t_{n} + j) for 0 ⩽ j ⩽ i and so the balls in T (t_{n} + j) are disjoint. By adapting the argument used in establishing (57), it is easily verified that

#V_{B}^{u}(t_{n} + i) ≥ \frac{1}{2} #G_{B}^{u}(t_{n} + i).  (62)

Now let

K(t_{n} + i, B) := ⋃_{α∈V_{B}^{u}(t_{n}+i)} B(R_{α}, ψ(u_{t_{n}+i})) .

This completes the inductive step and the construction of the local level K(n, B) associated with B ∈ K(n − 1). Clearly, for 0 ≤ i ≠ j ≤ k_{n}(B) we have that

K(t_{n} + i, B) ∩ K(t_{n} + j, B) = ∅ .

Furthermore, by construction K(n, B) is contained in B. Therefore, K(n) ⊂ K(n − 1). The Cantor set K_{η} is defined as

K_{η} := ⋂_{n=1}^{∞} K(n) .

Strictly speaking, K_{η} \ R ⊂ Λ(R, β, Ψ) and not K_{η} ⊂ Λ(R, β, Ψ). However, this is irrelevant since R is countable and so H^{s}(K_{η} \ R) = H^{s}(K_{η}). Before constructing a measure on K_{η}, we state an important lemma. The proof is a simple consequence of (50), (57) and (62).

Lemma 10 (i) For 0 ⩽ i ⩽ k_{1},

#V_{I_{0}}^{u}(t_{1} + i) ψ(u_{t_{1}+i})^{s} ⩾ \frac{c_{1}|I_{0}|}{8} g(u_{t_{1}+i}) .

(ii) For n ⩾ 2, let B be a ball in K(n − 1). Then, for 0 ⩽ i ⩽ k_{n}(B)

#V_{B}(t_{n} + i) ψ(u_{t_{n}+i})^{s} ⩾ \frac{c_{1}|B|}{8} g(u_{t_{n}+i}) .

C.2.2. A measure on K_{η}

In this section, we construct a probability measure μ supported on K_{η} satisfying (49). Suppose n ≥ 2 and B ∈ K(n). For 1 ≤ m < n, let B_{m} denote the unique ball in K(m) containing the ball B. With this notation in mind we now define μ. For any B ∈ K(n), we attach a weight μ(B) defined recursively as follows: For n = 1,

μ(B) := \frac{r(B)^{s}}{∑_{B′∈K(1)} r(B′)^{s}}

and for n ⩾ 2,

μ(B) := \frac{r(B)^{s}}{∑_{B′∈K(n,B_{n−1})} r(B′)^{s}} × μ(B_{n−1}) .

This procedure defines inductively a mass on any ball appearing in the construction of K_{η}. In fact a lot more is true: The probability measure μ constructed above is supported on K_{η} and for any Borel subset F of I_{0}

μ(F) := μ(F ∩ K_{η}) = inf ∑_{B∈B} μ(B) ,

where the infimum is taken over all coverings B of F ∩ K_{η} by balls B ∈ {K(n) : n ∈ ℕ}.

For further details see [19, Prop. 1.7] . It remains to establish (49) for μ.

Measure of a ball in the Cantor construction. If B ∈ K(n) for some n ∈ ℕ, then

μ(B) := \frac{r(B)^{s}}{∑_{B′∈K(n,B_{n−1})} r(B′)^{s}} × μ (B_{n−1})

= \frac{r(B)^{s}}{∑_{B′∈K(1)} r(B′)^{s}} ∏_{m=1}^{n−1} \frac{r(B_{m})^{s}}{∑_{B′∈K(m+1,B_{m})} r(B′)^{s}} .  (63)

The product term is taken to be one when n = 1. To proceed we require the following lemma which gives us a lower bound on the terms in the denominator of the above expression.

Lemma 11

∑_{B∈K(1)} r(B)^{s} ⩾ \frac{η}{2} |I_{0}| and ∑_{B∈K(n,B_{n−1})} r(B)^{s} ⩾ r(B_{n−1})^{s} (n ⩾ 2).

Proof of Lemma 11. By Lemma 10, the choice of k_{1} (56) and ϖ (52) it follows that

∑_{B∈K(1)} r(B)^{s} = ∑_{i=0}^{k_{1}} #V_{I_{0}}^{u}(t_{1} + i) ψ(u_{t_{1}+i})^{s} ⩾ \frac{c_{1}|I_{0}|}{8} ∑_{i=0}^{k_{1}} g(u_{t_{1}+i}) > \frac{c_{1}|I_{0}|}{192} \frac{η}{ϖ} ⩾ \frac{η}{2} |I_{0}| .

The statement for n ≥ 2 follows in a similar fashion – use (60) rather than (56). ♠ In view of the above lemma, it now follows from (63) that for any ball B ∈ K(n)

μ(B) ⩽ \frac{2 r(B)^{s}}{|I_{o}| η} ≪ \frac{r(B)^{s}}{η} .  (64)

Measure of an arbitrary ball. The aim is to show that μ(A) ≪ r(A)^{s}/η for an arbitrary ball A with radius r(A) ≤ r_{o}. The measure μ is supported on K_{η}. Thus, without loss of generality we can assume that A ∩ K_{η} ≠ ∅; otherwise μ(A) = 0 and there is nothing to prove. We can also assume that for every n large enough A intersects at least two balls in K(n); since if B is the only ball in K(n) which has non–empty intersection with A, then in view of (64)

μ(A) ≤ μ(B) ≪ r(B)^{s}/η → 0 as n → ∞

(r(B) → 0 as n → ∞) and again there is nothing to prove. Thus we may assume that there exists an integer n ≥ 2 such that A intersects only one ball B̃ in K(n−1) and at least two balls from K(n). The case that A intersects two or more balls from the first level can be excluded by choosing r(A) sufficiently small. This follows from the fact that by construction balls in any one level are disjoint. Furthermore, we can assume that

r(A) < r(B̃) .

Otherwise, μ(A) ≤ μ(B̃) ≤ r(B̃)^{s}/η ≤ r(A)^{s}/η and we are done. Given that A only intersects the ball B̃ in K(n−1), the balls from level K(n) which intersect A must be contained in the local level

K(n, B̃) := ⋃_{i=0}^{k_{n}(B̃)} K(t_{n} + i, B̃) .

By construction, any ball B(R_{α}, ψ(u_{t_{n}+i})) in K(n, B̃) is contained in the ball B(R_{α}, hB̃(u_{t_{n}+i})). Thus A intersects at least one ball in T (t_{n} + i, B̃) for some 0 ≤ i ≤ k_{n}(B̃).

Let K(t_{n} + i′, B̃) be the first local sub-level associated with B̃ such that

K(t_{n} + i′, B̃) ∩ A ≠ ∅ .

Thus, A intersects at least one ball B(R_{α}, ψ(u_{t_{n}+i′})) from K(t_{n} + i′, B̃) and such balls are indeed the largest balls from the n’th level K(n) that intersect A. Clearly, A intersects at least one ball B_{∗} in T (t_{n} + i′, B̃). We now prove a trivial but crucial geometric lemma.

Lemma 12 For i ≥ i′, if A intersects B(R_{α}, ψ(u_{t_{n}+i})) ⊂ B(R_{α}, hB̃(u_{t_{n}+i})) ≠ B_{∗} then

r(A) > \frac{1}{2} ρ(u_{t_{n}+i}) .

Proof of Lemma 12. If i = i′ then as balls of radius ρ(u_{t_{n}+i′}) are disjoint we have that r(A) > ρ(u_{t_{n}+i′}). Assume that i > i′, then by construction B(R_{α}, ρ(u_{t_{n}+i})) ∩ B_{∗} = ∅. Hence, r(A) > ρ(u_{t_{n}+i}) − ψ(u_{t_{n}+i}) > \frac{1}{2}ρ(u_{t_{n}+i}) – see (51). ♠

In view of the definition of i′ and (64), we have that

μ(A) ≤ ∑_{i = i′}^{k_{n}(B̃)} ∑_{\begin{matrix}α∈ V u ; B̃(t_{n}+i) : \\ B(R_{α},ψ(u_{tn+i}))∩A≠∅ \end{matrix}} μ(B(R_{α}, ψ(u_{t_{n}+i})))

≤ \frac{2}{η |I_{0}|} ∑_{i = i′}^{k_{n}(B̃)} ψ(u_{t_{n}+i})^{s} ∑_{\begin{matrix}α∈V u ; B̃(t_{n}+i) : \\ B(R_{α},ψ(u_{tn+i}))∩A≠∅ \end{matrix}} 1 .  (65)

In view of Lemma 12, if A intersects some ball B(R_{α}, hB̃(t_{n} + i)) in T (t_{n} + i, B̃) then the ball B(R_{α}, ρ(u_{t_{n}+i})) which contains it is itself contained in the ball 5A. Let N_{i} denote the number of balls B(R_{α}, ρ(u_{t_{n}+i})) with α ∈ V u ; B̃(t_{n} + i) that can possibly intersect A. By construction these balls are disjoint. Thus, 2ρ(u_{t_{n}+i}) × N_{i} ≤ |5A| = 10 r(A). This implies, via (65) that

μ(A) ≤ \frac{2}{η |I_{0}|} ∑_{i = i′}^{k_{n}(B̃)} ψ(u_{t_{n}+i})^{s} N_{i} ≤ \frac{10}{η |I_{0}|} r(A) ∑_{i = 0}^{k_{n}(B̃)} g(u_{t_{n}+i}) .

By (60),

∑_{i=0}^{k_{n}(B)−1} g(u_{t_{n}+i}) ≤ \frac{r(B̃)^{s−1}}{24ϖ} ,

and by (59) together with the fact that g(u_{n}) < G^{∗} for all n

g(u_{t_{n}+k_{n}(B̃)}) < \frac{r(B̃)^{s−1}}{24ϖ} .

Hence

μ(A) ≪ \frac{1}{η} r(A) r(B̃)^{s−1}  (66)

However, r(A) < r(B̃) and s < 1. The desired inequality, namely (49) now readily follows. This completes the proof of Theorem 10 in the case that G is finite. ♠

C.3. Proof of Theorem 10: G = ∞

The proof of Theorem 10 in the case that G is infinite follows the same strategy as the proof when G is finite. However, to execute the strategy is far simpler than in the finite case.

C.3.1. The Cantor set K and the measure μ

We start by defining a Cantor set K which is dependent on a certain, strictly increasing sequence of natural numbers {t_{i} : i ∈ ℕ}. The main difference between this case and the previous case is that we do not need to consider sublevels.

The Cantor set K. Choose t_{1} sufficiently large so that the counting estimate (50) is valid for the set G_{I_{0}}^{u}(t_{1}) and define the first level K(1) of the Cantor set K as follows:

K(1) := ⋃_{α∈G_{I_{0}}^{u}(t_{1})} B(R_{α}, ψ(u_{t_{1}})) .

For n ≥ 2 we define the n’th level K(n) recursively as follows:

K(n) := ⋃_{B∈K(n−1)}^{∘} K(n, B) ,

where

K(n, B) := ⋃_{α∈G_{B}^{u}(t_{n})} B(R_{α}, ψ(u_{t_{n}}))

is the n’th local level associated with the ball B := B(R_{α}, ψ(u_{t_{n−1}})) ∈ K(n−1). Here t_{n} > t_{n−1} is chosen sufficiently large so that (50) is valid for any ball B in K(n − 1). By construction K(n, B) ⊂ B and so K(n) ⊂ K(n − 1). The Cantor set K is simply given by

K := ⋂_{n=1}^{∞} K(n) .

By construction, K \ R ⊂ Λ(R, β, Ψ) and since R is countable H^{s}(Λ(R, β, Ψ)) ≥ H^{s}(K).

The measure μ. Suppose n ≥ 2 and B ∈ K(n). For 1 ≤ m < n, let B_{m} denote the unique ball in K(m) containing B. For any B ∈ K(n), we attach a weight μ(B) defined recursively as follows:

For n = 1,

μ(B) := \frac{1}{#G_{I_{0}}^{u}(t_{1})}

and for n ⩾ 2,

μ(B) := \frac{1}{#G_{B_{n−1}}^{u}(t_{n})} × μ(B_{n−1}) .

By the definition of μ and the counting estimate (50), it follows that

μ(B) ≤ \frac{2}{|I_{0}|} c_{1}^{−n} ρ(u_{t_{n}}) × ∏_{m=1}^{n−1} \frac{ρ(u_{t_{m}})}{ψ(u_{t_{m}})} .  (67)

The product term is taken to be one when n = 1.

C.3.2. Completion of the proof

Fix η ≥ 1. Since G = ∞, the sequence {t_{i}} associated with the construction of the Cantor set K can clearly be chosen so that

η × \frac{2}{|I_{0}|} c_{1}^{−i} × ∏_{j=1}^{i−1} \frac{ρ(u_{t_{j}})}{ψ(u_{t_{j}})} ≤ g(u_{t_{i}}) .  (68)

The product term is one when i = 1. It now immediately follows from (67) that for any B ∈ K(n),

μ(B) ≤ r(B)^{s} \frac{2}{|I_{0}|} c_{1}^{−n} \frac{1}{g(u_{t_{n}})} × ∏_{m=1}^{n−1} \frac{ρ(u_{t_{m}})}{ψ(u_{t_{m}})} ≤ r(B)^{s}/η .

We now show that μ(A) ≪ r(A)^{s}/η where A is an arbitrary ball. The same reasoning as before enables us to assume that A ∩ K ≠ ∅ and that there exists an integer n ≥ 2 such that A intersects only one ball B̃ in K(n − 1) and at least two balls from K(n, B̃) ⊂ K(n). Thus,

ρ(u_{t_{n}}) ≤ r(A) ≤ r(B̃) := Ψ(u_{t_{n−1}}) .  (69)

The left hand side of (69) makes use of the fact that B(R_{α}, ψ(u_{t_{n}})) ⊂ B(R_{α}, ρ(u_{t_{n}})) and that the balls B(R_{α}, 2ρ(u_{t_{n}})) with α ∈ Gu ; B̃(t_{n}) are disjoint. A simple geometric argument yields that N := #{α ∈ Gu ; B̃(t_{n}) : B(R_{α}, ρ(u_{t_{n}})) ∩ A ≠ ∅} ≤ 3 r(A)/ρ(u_{t_{n}}) . In view of (67), (68),

(69) and the fact that s < 1, we obtain

μ(A) ≤ N μ(B(R_{α}, ψ(u_{t_{n}}))) ⩽ r(A) \frac{6}{|I_{0}|} c_{1}^{−n} ∏_{m=1}^{n−1} \frac{ρ(u_{t_{m}})}{ψ(u_{t_{m}})}

⩽ r(A)^{s} ψ(u_{t_{n−1}})^{1−s} \frac{6}{|I_{0}|} c_{1}^{−n} ∏_{m=1}^{n−1} \frac{ρ(u_{t_{m}})}{ψ(u_{t_{m}})}

⩽ r(A)^{s} \frac{6}{|I_{0}|} c_{1}^{−n} \frac{1}{g(u_{t_{n−1}})} ∏_{m=1}^{n−2} \frac{ρ(u_{t_{m}})}{ψ(u_{t_{m}})} ≤ 3 c^{−1} \frac{r(A)^{s}}{η} .

The upshot is that (49) is satisfied and thereby completes the proof. ♠

Acknowledgements. We would like to thank the referee for making many useful suggestions. In particular, it was the referees comments which lead us to the convergent statements for rational quadrics – thank you for sharing your insight. Previously, we had only obtained these statements for the unit circle.

As ever, SV would like to thank his old friend Bridget and his new friends Ayesha and Iona for bringing so much love and laughter into his life.

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Victor V. Beresnevich: Institute of Mathematics, Academy of Sciences of Belarus, 220072, Surganova 11, Minsk, Belarus. e-mail: beresnevich@im.bas-net.by

Detta Dickinson: Department of Mathematics, National University of Ireland, Maynooth, Co.Kildare, Ireland. e-mail: : ddickinson@maths.may.ie

Sanju L. Velani: Department of Mathematics, University of York, Heslington, York, YO10 5DD, England. e-mail: slv3@york.ac.uk

Appendix II: Sums of Two Squares Near Perfect Squares

R.C. Vaughan

A.1 The theorem

Let r(n) denote the number of representations of a number n as the sum of two squares of integers and let ψ : ℕ → ℝ be a non-negative decreasing function. We prove the following theorem.

Theorem A Let Q^{∗} denote the smallest integer with Q^{∗} > Q. Then for each real number Q and natural number N with N ⩽ Q^{3},

∑_{Q<q⩽2Q} ∑_{n}′r(n) = ∑_{Q<q⩽2Q} 4πqψ(q)

+ O(Q log Q + Q^{\frac{3}{2}} ψ(Q^{∗})^{\frac{1}{2}} (log Q)^{64} + Q^{2}ψ(Q^{∗})^{\frac{1}{2}} (log Q)^{64}N^{− \frac{1}{4}}

+ N^{\frac{3}{4}} (log N)^{3}Q^{\frac{1}{2}} ψ(Q^{∗}) + N^{\frac{1}{4}} (log N)Q^{\frac{1}{2}} ∑_{Q<q⩽2Q} ψ(q) + Q^{2}(log Q)^{3}N^{− \frac{1}{2}} )

where ∑′ indicates that the sum is over n with |q − \sqrt{n}| ⩽ ψ(q) and that any terms with |q − \sqrt{n}| = ψ(q) are counted with weight \frac{1}{2}.

When ψ(Q^{∗}) has the same order of magnitude as Q^{−1} ∑_{Q<q⩽2Q} ψ(q) and the sum ∑_{Q<q⩽2Q} ψ(q) is large, a good choice for N is

Q^{2} ( ∑_{Q<q⩽2Q} ψ(q))^{−1} .

This leads to the error estimate

≪ Q ( ∑_{Q<q⩽2Q} ψ(q))^{\frac{3}{4}} (log Q)^{64} .

Then the main term will dominate provided that ∑_{Q<q⩽2Q} ψ(q) is large compared with (log Q)^{256}. A concomitant remark pertains if the theorem is averaged over Q with, say R < Q ⩽ 2R. It may well be possible to replace the (log Q)^{64} in the error term by a smaller power of log Q, but that some power of a logarithm has to be present follows from either of the observations that

∑_{q⩽Q} r(q^{2}) ∼ \frac{4}{π} Q log Q

(see (75) below) and

∑_{q⩽Q} r(q^{2} + 1) ∼ \frac{12}{π}Q log Q.

A.2 Proof of Theorem

Let R(x) = ∑_{1⩽n⩽x} r(n), Δ(x) = R(x) − πx, and Δ_{0}(x) = Δ(x) when x ∉ ℕ and Δ_{0}(x) = Δ(x) − \frac{1}{2}r(x) when x ∈ ℕ. Then our motivation is the formula of Hardy [1, pg 265] which, for real x ⩾ x_{0}, we restate as

Δ_{0}(x) = −1 + \sqrt{x} ∑_{n=1}^{∞} r(n)n^{−1/2} J_{1} (2π\sqrt{nx})

where J_{1} denotes the usual Bessel function. However the convergence is only conditional and we require a form of this in which the tail of the infinite series is more readily accessible.

By Theorem 1 and Lemma 2 of [2] for any δ with 0 < δ < 1 and provided that x ⩾ x_{0}(δ) and N > N_{0}(δ) we have

Δ_{0}(x) = −1 + \sqrt{x} ∑_{1⩽n⩽N} \frac{r(n)}{n^{\frac{1}{2}}} J_{1} (2π\sqrt{nx})

− x^{\frac{5}{4}} ∑_{x(1−δ)<n<x(1+δ)} \frac{r(n)}{πn^{\frac{5}{4}}} sgn(\sqrt{\frac{n}{x}} − 1) ∫_{2π|\sqrt{n}−\sqrt{x}|\sqrt{N}}^{∞} \frac{sin α}{α} dα
+ O ((xN)^{− \frac{1}{4}} + x^{\frac{1}{4}} N^{− \frac{5}{12}} )

where we have used Δ(x) ≪ x^{1/3} of [3] and sgn(u) is −1, 0 or 1 according as u < 0, u = 0 or u > 0. A standard estimate for J_{1} [4, pg 199] gives for α > α_{0}

J_{1}(2πα) = −\frac{1}{π} α^{−1/2} cos (2πα + \frac{π}{4} ) + O (α^{−3/2}) .

For convenience we put

S(x) = x^{1/4} ∑_{1⩽n⩽N} r(n)n^{−3/4} cos (2π\sqrt{nx} + \frac{π}{4} )

and

E(x) = x^{\frac{5}{4}} ∑_{x(1−δ)<n<x(1+δ)} \frac{r(n)}{n^{\frac{5}{4}}} sgn(\sqrt{\frac{n}{x}} − 1) ∫_{2π|\sqrt{n}−\sqrt{x}|\sqrt{N}}^{∞} \frac{sin α}{α} dα

so that

Δ_{0}(x) = −1 − π^{−1}(S(x) + E(x)) + O (x^{− \frac{1}{4}} + x^{\frac{1}{4}} N^{− \frac{5}{12}} )
= −1 − π^{−1}(S(x) + E(x)) + O ((x/N)^{\frac{1}{2}} )

since x^{− \frac{1}{4}} + x^{\frac{1}{4}} N^{− \frac{5}{12}} ≪ (x/N)^{\frac{1}{2}} whenever N ≪ x^{3/2}. For x_{0} ⩽ x ⩽ y we have

S(y) − S(x) = ∑_{1⩽n⩽N} \frac{r(n)}{n^{3/4}} ∫_{x}^{y} ℜ ((\frac{1}{4}u^{−3/4} + πiu^{−1/4}n^{1/2}) e (\sqrt{nu} + \frac{1}{8})) du.

The contribution to S(y) − S(x) from the

\frac{1}{4}u^{−3/4}e (\sqrt{nu} + \frac{1}{8})

part of the integrand is ≪ x^{−3/4}N^{1/4}(y − x). Here we have used partial summation and the fact that r(n) is on average π. We shall do this several times hereafter without comment.

To prove the theorem we may suppose that Q > Q_{0}. In particular Q_{0} can be chosen so that q − ψ(q) > 2 whenever q > Q. Thus, when Q < q ⩽ 2Q,

Δ_{0}((q + ψ(q))^{2}) − Δ_{0}((q − ψ(q))^{2}) = −\frac{1}{π} T (q, N)
+ O (N^{\frac{1}{4}} q^{− \frac{1}{2}} ψ(q) + E_{+}(q, N) + E_{−}(q, N) + qN^{− \frac{1}{2}} )  (70)

where

T (q, N) = ∑_{1⩽n⩽N} \frac{r(n)}{n^{1/4}} ∫_{(q−ψ(q))^{2}}^{(q+ψ(q))^{2}} ℜ (πiu^{−1/4}e (\sqrt{nu} + \frac{1}{8})) du

and

E_{±}(q, N) = ∑_{\frac{1}{2} Q^{2}<n<8Q^{2}} r(n) min (1, \frac{1}{|\sqrt{n} − (q ± ψ(q)|\sqrt{N}} ) .

In the integral in T (q, N) we make the change of variables, u = (q + t)^{2}, so that

T (q, N) = ∑_{1⩽n⩽N} \frac{r(n)}{n^{1/4}} ∫_{−ψ(q)}^{ψ(q)} ℜ (2πi(q + t)^{1/2}e ((q + t)\sqrt{n} + \frac{1}{8})) dt .

The factor (q + t)^{1/2} in the integrand is q^{1/2} + O(|t|q^{−1/2}) and so

T (q, N) = U(q, N) + O (q^{− \frac{1}{2}} ψ(q)^{2}N^{3/4}) ,  (71)

where

U(q, N) = ∑_{1⩽n⩽N} \frac{r(n)}{n^{1/4}} ∫_{−ψ(q)}^{ψ(q)} ℜ (2πiq^{1/2}e ((q + t)\sqrt{n} + \frac{1}{8})) dt .

Collecting together the estimates (70) and (71) we have

Δ_{0} ((q + ψ(q))^{2}) − Δ_{0}((q − ψ(q))^{2}) = −\frac{1}{π} U(q, N)
+ O (N^{\frac{3}{4}} q^{− \frac{1}{2}} ψ(q)^{2} + N^{\frac{1}{4}} q^{− \frac{1}{2}} ψ(q) + E_{−}(q, N) + E_{+}(q, N) + qN^{− \frac{1}{2}} ) . (72)

Let Q^{∗} denote the smallest integer q with q > Q. Then

∑_{Q<q⩽2Q}U(q, n) = ∑_{1⩽n⩽N} \frac{r(n)}{n^{1/4}} ∫_{−ψ(Q^{∗})}^{ψ(Q^{∗})} ℜ (2πie (t\sqrt{n} + \frac{1}{8}) ∑_{Q<q⩽2Q:ψ(q)⩾|t|} q^{\frac{1}{2}} e(q\sqrt{n})) dt .

We also have

∑_{Q<q⩽2Q} ∑_{\frac{1}{2} Q^{2}<n<8Q^{2}} r(n) min (1, \frac{1}{|\sqrt{n} − (q ± ψ(q)|\sqrt{N}} )

≪ ∑_{Q<q⩽2Q} ∑_{\frac{1}{2} Q^{2}<n<8Q^{2}} r(n) min (1, \frac{Q}{|n − (q ± ψ(q))^{2}|\sqrt{N}} )

≪ ∑_{Q<q⩽2Q} ∑_{−q^{2}<h⩽8Q^{2}} r(q^{2} + h) min (1, \frac{Q}{|h ∓ 2qψ(q) − ψ(q)|\sqrt{N}} ) .

and

∑_{Q<q⩽2Q} ∑_{\begin{matrix}4qψ(q)+2ψ(q)^{2}<|h|⩽8Q^{2} \\ h>−q^{2} \end{matrix}} r(q^{2} + h) min (1, \frac{Q}{|h ∓ 2qψ(q) − ψ(q)|\sqrt{N}} )

≪ ∑_{Q<q⩽2Q} ∑_{\begin{matrix}4qψ(q)+2ψ(q)^{2}<|h|⩽8Q^{2} \\ h>−q^{2} \end{matrix}} d(q^{2} + h) min (1, \frac{Q}{|h|\sqrt{N}} ) .

Here we observe that

r(n) ⩽ 4d(n) ⩽ 8 ∑_{\begin{matrix}l|n \\ l⩽\sqrt{n} \end{matrix}} 1 .

Below we state a bound for the number of solutions of a quadratic congruence which we use several times over and which is readily established using elementary facts about such congruences.

Lemma 13 Suppose that m ∈ ℕ, h ∈ ℤ and define d_{1} and d_{2} uniquely by taking (m, h) = d_{1}d_{2}^{2} where d_{1} is square free. Further let ρ(m; h) denote the number of solutions of the congruence y^{2} + h ≡ 0 (mod m) in y modulo m. Then

ρ(m; h) ⩽ 2d_{2}d (\frac{m}{d_{1}d_{2}^{2}} ) .

By Lemma 13

∑_{Q<q⩽2Q}  ∑_{\begin{matrix}4qψ(q)+2ψ(q)^{2}<|h|⩽8Q^{2} \\ h>−q^{2} \end{matrix}} d(q^{2} + h) min (1, \frac{Q}{|h|\sqrt{N}} )

≪ ∑_{Q<q⩽2Q} ∑_{4qψ(q)+2ψ(q)^{2}<|h|⩽8Q^{2}} ∑_{m|q^{2}+h,m≪q} min (1, \frac{Q}{|h|\sqrt{N}} )

≪ ∑_{0<d_{1}d_{2}^{2}|j|≪Q} ∑_{l≪Q/d_{1}d_{2}^{2}} \frac{Q^{2}d(l)}{d_{1}^{2}d_{2}^{3}l|j|\sqrt{N}} ≪ Q^{2}N^{− \frac{1}{2}} (log Q)^{3}.

Hence, by (72),

∑_{Q<q⩽2Q} (Δ_{0}((q + ψ(q))^{2}) − Δ_{0}((q − ψ(q))^{2})) = V (Q, N)

+ O(N^{\frac{3}{4}} ∑_{Q<q⩽2Q} q^{− \frac{1}{2}} ψ(q)^{2} + N^{\frac{1}{4}} ∑_{Q<q⩽2Q} q^{− \frac{1}{2}} ψ(q)

+ Q^{2}N^{− \frac{1}{2}} (log Q)^{3} + F_{−}(Q, N) + F_{+}(Q, N))  (73)

where

V (Q, N) = ∑_{1⩽n⩽N} \frac{r(n)}{n^{1/4}}ℑ ∫_{−ψ(Q^{∗})}^{ψ(Q^{∗})} 2e (t\sqrt{n} + \frac{1}{8}) ∑_{\begin{matrix}Q<q⩽2Q \\ ψ(q)⩾|t| \end{matrix}} q^{1/2}e (q\sqrt{n}) dt

and

F_{±}(Q, N) = ∑_{Q<q⩽2Q} ∑_{\begin{matrix}|h|⩽4qψ(q)+2ψ(q)^{2} \\ h>−q^{2} \end{matrix}} r(q^{2} + h) min (1, \frac{Q}{|h ∓ 2qψ(q) − ψ(q)^{2}|\sqrt{N}} ) . (74)

When q > Q, let

W (q) = ∑_{r=Q^{∗}}^{q} e(r\sqrt{n})

and suppose Q^{∗} ⩽ m ⩽ 2Q. Then

∑_{q=Q^{∗}}^{m}q^{\frac{1}{2}} e (q\sqrt{n}) = ∑_{q=Q^{∗}}^{m}q^{\frac{1}{2}} (W (q) − W (q − 1)) = − ∑_{q=Q^{∗}}^{m−1} ((q + 1)^{\frac{1}{2}} − q^{\frac{1}{2}} ) W (q) + m^{\frac{1}{2}} W (m)

≪ Q^{1/2} min (m − Q^{∗} + 1, \frac{1}{∥\sqrt{n}∥}) = Q^{1/2} min ( ∑_{q=Q^{∗}}^{m} 1, \frac{1}{∥\sqrt{n}∥}) .

We have

∫_{−ψ(Q^{∗})}^{ψ(Q^{∗})} min ( ∑_{\begin{matrix}Q<q⩽2Q \\ ψ(q)⩾|t| \end{matrix}} 1, \frac{1}{∥\sqrt{n}∥}) dt ≪ min (∫_{−ψ(Q^{∗})}^{ψ(Q^{∗})} ∑_{\begin{matrix}Q<q⩽2Q \\ ψ(q)⩾|t| \end{matrix}} dt, \frac{ψ(Q^{∗})}{∥\sqrt{n}∥}) .

Therefore,

V (Q, N) ≪ Q^{1/2} ∑_{1⩽n⩽N} \frac{r(n)}{n^{1/4}} min ( ∑_{Q<q⩽2Q} ψ(q), \frac{ψ(Q^{∗})}{∥\sqrt{n}∥}) .

Suppose that 1 ⩽ m ⩽ \sqrt{N} + \frac{1}{2}, and consider those n with (m − \frac{1}{2})^{2} < n ⩽ (m + \frac{1}{2})^{2}. Then ∥\sqrt{n}∥ = |\sqrt{n} − m| = \frac{|n−m^{2}|}{\sqrt{n}+m} ≫ \frac{|n−m^{2}|}{m}. Hence, when m > 1,

∑_{(m−1/2)^{2}<n⩽(m+1/2)^{2}} \frac{r(n)}{n^{1/4}} min ( ∑_{Q<q⩽2Q} ψ(q), \frac{ψ(Q^{∗})}{∥\sqrt{n}∥})

≪ \frac{r(m^{2})}{m^{\frac{1}{2}}} ∑_{Q<q⩽2Q} ψ(q) + m^{\frac{1}{2}} ∑_{0<|h|⩽m} \frac{r(m^{2} + h)}{|h|}ψ(Q^{∗}) .

The Dirichlet series generating function for r(m^{2}) is

4(1 + 2^{−s})^{−1}ζ(s)^{2}L(s)ζ(2s)^{−1} ,

where L(s) is the Dirichlet L-function formed from the non-trivial character modulo 4. Thus

∑_{m⩽M} r(m^{2}) ∼ \frac{4}{π} M log M  (75)

and hence

∑_{m⩽M} \frac{r(m^{2})}{m^{\frac{1}{2}}} ≪ M^{\frac{1}{2}} log M .

As in the analysis of E_{±} above we have

∑_{2⩽m⩽M} ∑_{0<|h|⩽m} \frac{r(m^{2} + h)}{|h|} ≪ M(log M)^{3} .

Hence

V (Q, N) ≪ N^{\frac{1}{4}} (log N)Q^{\frac{1}{2}} ∑_{Q<q⩽2Q} ψ(q) + N^{3/4}(log N)^{3}Q^{\frac{1}{2}} ψ(Q^{∗}) .

Hence, assuming N ⩽ Q^{3}, by (73),

∑_{Q<q⩽2Q} (Δ_{0}((q + ψ(q))^{2}) − Δ_{0}((q − ψ(q))^{2}))
≪ N^{\frac{1}{4}} (log N)Q^{\frac{1}{2}} ∑_{Q<q⩽2Q} ψ(q) + N^{\frac{3}{4}} (log N)^{3}Q^{\frac{1}{2}} ψ(Q^{∗})
+ Q^{2}N^{− \frac{1}{2}} (log Q)^{3} + F_{−}(Q, N) + F_{+}(Q, N) . (76)

We now turn our attention to F_{±}. Were the factor r(q^{2} + h) not to be present this would be a routine matter. The natural way to remove it is to consider an application of the Cauchy-Schwarz inequality. However one is then dependent on being able to bound r(n)^{2}, or d(n)^{2} in terms of the divisors of n of order of magnitude at most \sqrt{n}. This is readily effected by an application of a combinatorial lemma.

Lemma 14 Let n ∈ ℕ. Then there is a divisor m of n such that m ⩽ \sqrt{n} and d(n) ⩽ max(2, d(m)^{3}).

Proof. The conclusion follows at once when n has a prime factor p with p > \sqrt{n}. Otherwise choose a sequence \{m_{j}\} as follows. Let m_{1} be the largest divisor of n not exceeding \sqrt{n}. Then given m_{1}, m_{2}, …, m_{j} with m_{1} … m_{j}|n and no m_{k} exceeding \sqrt{n} choose m_{j+1} to be the largest divisor of n/(m_{1} … m_{j}) not exceeding \sqrt{n}. It follows that m_{4} = 1 since otherwise we would have m_{1} m_{2} > \sqrt{n} and m_{3} m_{4} > \sqrt{n}. Hence n = m_{1} m_{2} m_{3} and d(n) ⩽ d(m_{j})^{3} for some j. ♠

By Lemma 14,

∑_{Q<q⩽2Q} ∑_{\begin{matrix}0<|h|⩽4qψ(q)+2ψ(q)^{2} \\ h>−q^{2} \end{matrix}} d(q^{2} + h)^{2}

≪ ∑_{0<|h|⩽4Qψ(Q^{∗})+2ψ(Q^{∗})^{2}}  ∑_{Q<q⩽2Q}  ∑_{m|q^{2}+h, m≪Q} d(m)^{6}

and, by Lemma 13, this is

≪ ∑_{0<d_{1}d_{2}^{2}|j|⩽4Qψ(Q^{∗})+2ψ(Q^{∗})^{2}}  ∑_{l≪Q/(d_{1}d_{2}^{2})} d(ld_{1}d_{2}^{2})^{6} \frac{Q}{ld_{1}d_{2}} d(l)

≪ Q^{2} ψ(Q^{∗}) (log Q)^{128} .

We also have

∑_{Q<q⩽2Q} ∑_{|h|⩽4qψ(q)+2ψ(q)^{2}} min (1, \frac{Q^{2}}{|h ∓ 2qψ(q) − ψ(q)^{2}|^{2}N} ) ≪ Q + Q^{2}N^{− \frac{1}{2}} .

Hence, by (75) and the Cauchy-Schwarz inequality,

F_{±}(Q, N) ≪ Q log Q + Q^{\frac{3}{2}} ψ(Q^{∗})^{\frac{1}{2}} (log Q)^{64} + Q^{2}ψ(Q^{∗})^{\frac{1}{2}} (log Q)^{64}N^{− \frac{1}{4}} ,

and the theorem follows from (76). ♠

References

[1] G.H. Hardy: On the expression of a number as the sum of two squares. Quarterly Journal of Mathematics 46 (1915) 263–283.

[2] G.H. Hardy: The lattice points of a circle. Proceedings of the Royal Society, A 105 (1925) 623–635.

[3] W. Sierpiński: O pewnem zagadnieniu z rachunku funkcyj asymptotycznych. Prace mat.- fiz. 17 (1906) 77–118.

[4] G.N. Watson: A treatise on the theory of Bessel functions. second ed., Cambridge University Press (1944).

RCV: Department of Mathematics, McAllister Building, Pennsylvania State University, University Park, PA 16802, U.S.A.