A note on simultaneous Diophantine approximation on planar curves

Victor Beresnevich^{∗}

Minsk

Sanju Velani^{†}

York

For Iona and Ayesha on N^{o}3

Abstract

Let S_{n}(ψ_{1}, …, ψ_{n}) denote the set of simultaneously (ψ_{1}, …, ψ_{n})–approximable points in ℝ^{n} and S_{n}^{∗}(ψ) denote the set of multiplicatively ψ–approximable points in ℝ^{n}. Let M be a manifold in ℝ^{n}. The aim is to develop a metric theory for the sets M∩S_{n}(ψ_{1}, …, ψ_{n}) and M∩S_{n}^{∗}(ψ) analogous to the classical theory in which M is simply ℝ^{n}. In this note, we mainly restrict our attention to the case that M is a planar curve C. A complete Hausdorff dimension theory is established for the sets C ∩ S_{2}(ψ_{1}, ψ_{2}) and C ∩ S_{2}^{∗}(ψ). A divergent Khintchine type result is obtained for C ∩ S_{2}(ψ_{1}, ψ_{2}); i.e. if a certain sum diverges then the one–dimensional Lebesgue measure on C of C ∩ S_{2}(ψ_{1}, ψ_{2}) is full. Furthermore, in the case that C is a rational quadric the convergent Khintchine type result is obtained for both types of approximation. Our results for C ∩ S_{2}(ψ_{1}, ψ_{2}) naturally generalize the dimension and Lebesgue measure statements of [2]. Within the multiplicative framework, our results for C ∩ S_{2}^{∗}(ψ) constitute the first of their type.

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

Keywords and phrases: Metric Diophantine approximation, Khintchine type theorems, Hausdorff dimension, Multiplicative approximation

∗Research supported by EPSRC Grant R90727/01 †Royal Society University Research Fellow

1 Introduction

1.1 Background: two types of simultaneous approximation

Throughout ψ : ℝ^{+} → ℝ^{+} will denote a real, positive decreasing function and will be referred to as an approximating function. Given approximating functions ψ_{1}, …, ψ_{n}, a point y = (y_{1}, …, y_{n}) ∈ ℝ^{n} is called simultaneously (ψ_{1}, …, ψ_{n})–approximable if there are infinitely many q ∈ ℕ such that

∥qy_{i}∥ < ψ_{i}(q) 1 ⩽ i ⩽ n

where ∥x∥ = min{|x − m| : m ∈ ℤ}. In the case ψ_{i} : h → h^{−v_{i}} with v_{i} > 0, the point y is said to be simultaneously (v_{1}, …, v_{n})–approximable. The set of simultaneously (ψ_{1}, …, ψ_{n})– approximable points in ℝ^{n} will be denoted by S_{n}(ψ_{1}, …, ψ_{n}) and similarly S_{n}(v_{1}, …, v_{n}) will denote the set of simultaneously (v_{1}, …, v_{n})–approximable points in ℝ^{n}. Geometrically, y ∈ S_{n}(ψ_{1}, …, ψ_{n}) if it lies in infinitely many n–dimensional ‘rectangular’ regions centred at rational points with ‘size’ determined by ψ_{1}, …, ψ_{n}.

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

∏_{i=1}^{n} ∥qy_{i}∥ < ψ(q) .

In the case ψ : h → h^{−v} with v > 0 the point y is said to be multiplicatively v–approximable. The set of multiplicatively ψ–approximable points in ℝ^{n} will be denoted by S_{n}^{∗}(ψ) and similarly S_{n}^{∗}(v) will denote the set of multiplicatively v–approximable points in ℝ^{n}. In this multiplicative setup, y ∈ S_{n}^{∗}(ψ) if it lies in infinitely many n–dimensional ‘hyperbolic’ regions centred at rational points with ‘size’ determined by ψ.

It is readily verified that

S_{n}(ψ_{1}, …, ψ_{n}) ⊂ S_{n}^{∗}(ψ) whenever ψ ⩾ ψ_{1} ⋯ ψ_{n},  (1)
S_{n}(v_{1}, …, v_{n}) ⊂ S_{n}^{∗}(v) whenever v ⩽ v_{1} + ⋯ + v_{n}.  (2)

Also, in view of Minkowski’s linear forms theorem which gives rise to a general n–dimensional version of Dirichlet’s theorem,

S_{n}(v_{1}, …, v_{n}) = ℝ^{n}  if  v_{1} + ⋯ + v_{n} ≤ 1 .  (3)

This together with (2) implies that

S_{n}^{∗}(v) = ℝ^{n}  if  v ≤ 1 .  (4)

The Lebesgue theory. The following key results provide beautiful and simple criteria for the ‘size’ of the sets S_{n}(ψ_{1}, …, ψ_{n}) and S_{n}^{∗}(ψ) expressed in terms of n–dimensional Lebesgue measure | |_{ℝ^{n}}. The first is due to Khintchine [10] and the second is due to Gallagher [7].

Theorem K (1926). Let ψ be an approximating function. Then

|S_{n}(ψ_{1}, …, ψ_{n})|_{ℝ^{n}} = \{ \begin{matrix}ZERO & if ∑ ψ_{1}(h) … ψ_{n}(h) < ∞ \\ FULL & if ∑ ψ_{1}(h) … ψ_{n}(h) = ∞ \end{matrix} .

This theorem is a generalization of Khintchine’s 1924 result which deals with the special case ψ_{1} = ψ_{2} = ⋯ = ψ_{n}.

Theorem G (1962). Let ψ be an approximating function. Then

|S_{n}^{∗}(ψ)|_{ℝ^{n}} = \{ \begin{matrix}ZERO & if ∑ ψ(h)^{n} (log h)^{n−1} < ∞ \\ FULL & if ∑ ψ(h)^{n} (log h)^{n−1} = ∞ \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 sets in question satisfy a ‘zerofull’ law. The divergence parts of the above statements constitute the main substance of the theorems. The convergence parts are a simple consequence of the Borel-Cantelli lemma from probability theory. Trivially, the convergence parts imply that

|S_{n}(v_{1}, …, v_{n})|_{ℝ^{n}} = 0  if  v_{1} + ⋯ + v_{n} > 1

and

|S_{n}^{∗}(v)|_{ℝ^{n}} = 0 if v > 1 .

Note that the former statement is in fact a consequence of the latter and (2). In the case that the set in question is of Lebesgue measure zero, a more delicate attribute of the ‘size’ of the set is its Hausdorff measure and dimension. In this article we shall only be concerned with the dimension theory. The Hausdorff dimension of a set X ∈ ℝ^{n} is defined as follows. For ρ > 0, a countable collection {B_{i}} of Euclidean balls in ℝ^{n} with diameter diam(B_{i}) ≤ ρ for each i such that X ⊂ ⋃_{i} B_{i} is called a ρ-cover for X. Let s be a non-negative number and define H_{ρ}^{s}(X) = inf {∑_{i} diam(B_{i})^{s} : {B_{i}} is a ρ−cover of X}, where the infimum is taken over all possible ρ-covers of X. The Hausdorff dimension dim X of X is defined by infimum over s for which sup_{ρ>0} H_{ρ}^{s}(X) is zero.

The dimension theory. The following relatively recent results provide exact formulae for the ‘size’ of the sets S_{n}(v_{1}, …, v_{n}) and S_{n}^{∗}(v) expressed in terms of Hausdorff dimension. The first is due to Rynne [11] and the second is due to Bovey & Dodson [6].

Theorem R (1996). Let v_{1} ≥ v_{2} ≥ ⋯ ≥ v_{n} and v_{1} + v_{2} + ⋯ + v_{n} ≥ 1. Then

dim S_{n}(v_{1}, …, v_{n}) = min_{1≤k≤n} \{\frac{n + 1 + ∑_{i=k}^{n}(v_{k} − v_{i})}{1 + v_{k}}\} .

In the case v_{1} = v_{2} = ⋯ = v_{n}, the above statement reduces to the classical Jarník– Besicovitch theorem.

Theorem BD (1978). Let v ≥ 1. Then

dim S_{n}^{∗}(v) = n − 1 + \frac{2}{v + 1} .

1.2 Simultaneous approximation restricted to manifolds

Let M be a manifold in ℝ^{n}. In short, the aim is to develop a metric theory for the sets M ∩ S_{n}(ψ_{1}, …, ψ_{n}) and M ∩ S_{n}^{∗}(ψ) analogous to that described above in which M is simply ℝ^{n}. The fact that the points y of interest consist of dependent variables, reflecting the fact that y ∈ M introduces major difficulties in attempting to describe the measure theoretic structure of either set. This is true even in the specific case that M is a planar curve – the main subject of this article.

In order to make any reasonable progress it is not unreasonable to assume that the manifolds M under consideration are non-degenerate. Essentially, these are smooth submanifolds 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.

Trivially, if the dimension dim M of the manifold M in ℝ^{n} is strictly less than n then |M|_{ℝ^{n}} = 0. Thus, in attempting to develop a Lebesgue theory for the sets M∩S_{n}(ψ_{1}, …, ψ_{n}) and M ∩ S_{n}^{∗}(ψ) it is natural to use the induced Lebesgue measure | . |_{M} on M.

In 1998, D. Kleinbock & G. Margulis [9] proved the Baker-Sprindžuk conjecture:

Theorem KM (1998) Let M be a non-degenerate manifold in ℝ^{n}. Then

|M ∩ S_{n}^{∗}(v)|_{M} = 0  if v > 1 .  (5)

By inclusion (2), Theorem KM implies that for any non-degenerate manifold

|M ∩ S_{n}(v_{1}, …, v_{n})|_{M} = 0  if v_{1} + … + v_{n} > 1 . ^{1}  (6)

Also, note that in view of (3) and (4) both (5) and (6) are sharp. The first significant ‘clear cut’ statement was for planar curves. In 1964, Schmidt [12] established (6) in the case that M is a C^{(3)} non-degenerate planar curve and v_{1} = v_{2}.

The result of Kleinbock & Margulis gives some hope of developing a general metric theory for simultaneous approximation restricted to manifolds, analogous to that described in §1.1. As stepping stones, it is natural to consider the following explicit problems which ask for refinements of the measure zero statement of Kleinbock & Margulis.

Problem S1 : Given a non-degenerate manifold M ⊂ ℝ^{n} and v > 1 (respectively v_{1} + … + v_{n} > 1), what is the Hausdorff dimension of M ∩ S_{n}^{∗}(v) (respectively M ∩ S_{n}(v_{1}, …, v_{n}) ) ?

1When v_{1} = … = v_{n} this statement verifies the conjecture of Sprindžuk which was stated for analytic manifolds only.

Problem S2 : Given a non-degenerate manifold M ⊂ ℝ^{n} and an approximating function ψ (respectively ψ_{1}, …, ψ_{n}), what is the weakest condition under which M∩S^{∗}(ψ) (respectively M ∩ S_{n}(v_{1}, …, v_{n}) ) is of Lebesgue measure zero?

Problem S1 motivates the dimension theory for simultaneous approximation restricted to manifolds whilst Problem S2 motivates the convergent aspects of the Lebesgue theory. A priori, convergent statements are usually easier to establish than their divergent counterparts.

Until recently, the existing metric theory for simultaneous approximation restricted to manifolds was rather ad-hoc – see [3] for an account. Even in the simplest geometric and arithmetic situation in which the manifold is a genuine curve in ℝ^{2} the above problems seemed to have been impenetrable. However, in [3] we made significant progress towards developing a complete metric theory for the sets M ∩ S_{2}(ψ_{1}, ψ_{2}) with ψ_{1} = ψ_{2} and M a non-degenerate planar curve^{2}. In this paper we study the general simultaneous settings given to us by the above problems. This therefore includes the multiplicative setup. As in [3], we will mainly direct our efforts towards the case that the manifold M is a planar curve C. Thus, dim M = 1 and n = 2 in the above problems.

1.3 Statement of results

1.3.1 The Lebesgue theory

Theorem 1 Let ψ_{1}, ψ_{2} be approximating functions and let C be a C^{(3)} non-degenerate planar curve. Then

|C ∩ S_{2}(ψ_{1}, ψ_{2})|_{C} = FULL  if  ∑_{h=1}^{∞} ψ_{1}(h)ψ_{2}(h) = ∞ .

The next theorem shows that the above result is best possible. We establish the complementary ‘convergence result’ for a class Q of non-degenerate rational quadrics. A planar curve C is in Q if it is the image of either the unit circle C_{1} := {(x_{1}, x_{2}) ∈ ℝ^{2} : x_{1}^{2} + x_{2}^{2} = 1}, the parabola {(x_{1}, x_{2}) ∈ ℝ^{2} : x_{2} = x_{1}^{2}} or the hyperbola {(x_{1}, x_{2}) ∈ ℝ^{2} : x_{1}^{2} − x_{2}^{2} = 1} under a rational affine transformation of the plane.

Theorem 2 Let ψ_{1}, ψ_{2} be approximating functions and C ∈ Q. Then

|C ∩ S_{2}(ψ_{1}, ψ_{2})|_{C} = \{ \begin{matrix}ZERO & if ∑_{h=1}^{∞} ψ_{1}(h)ψ_{2}(h) < ∞ , \\ FULL & if ∑_{h=1}^{∞} ψ_{1}(h)ψ_{2}(h) = ∞ . \end{matrix}  (7)

These theorems are a generalization of the results in [3] which deal with the situation ψ_{1} = ψ_{2}. The next theorem is concerned with the multiplicative Lebesgue theory and is a refinement of Theorem KM for manifolds in Q.

2Note that in the case the manifold 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.

Theorem 3 Let ψ be an approximating function and C ∈ Q. Then

|C ∩ S_{2}^{∗}(ψ)|_{C} = 0  if  ∑_{h=1}^{∞} ψ(h) log h < ∞ .  (8)

1.3.2 The dimension theory

Regarding Problem S1, for planar curves we are able to give a complete description for either form of simultaneous approximation.

Theorem 4 Let f ∈ C^{(3)}(I_{0}), where I_{0} is an interval and C_{f} := {(x, f(x)) : x ∈ I_{0}}. Let v_{1} and v_{2} be positive numbers such that 0 < min(v_{1}, v_{2}) < 1 and v_{1} + v_{2} ≥ 1. Assume that

dim \{x ∈ I_{0} : f′′(x) = 0\} ⩽ \frac{2 − min(v_{1}, v_{2})}{1 + max(v_{1}, v_{2})} .

Then

dim C_{f} ∩ S_{2}(v_{1}, v_{2}) = \frac{2 − min(v_{1}, v_{2})}{1 + max(v_{1}, v_{2})} .  (9)

In the case v_{1} = v_{2}, this theorem generalizes the dimension results of [3]. Our next result is a general n–dimensional statement concerning Lipshitz manifolds; i.e. manifolds for which there exists an atlas of Lipshitz maps.

Theorem 5 Let M be an arbitrary Lipshitz manifold in ℝ^{n} of dimension dim M. Then

dim M ∩ S_{n}^{∗}(v) ⩾ dim M − 1 + \frac{2}{1 + v}  if  v ≥ 1 .  (10)

In essence, the above theorem indicates that the lower bound for the Hausdorff dimension in the general multiplicative setup reduces to a one dimensional problem. For clarification of this remark, see (43) in §6.2.

We conjecture that for manifolds in ℝ^{n} which are non-degenerate everywhere except possibly on a set of dimension at most dim M−1+2/(1+v), the lower bound given by Theorem 5 is in fact exact. The following result verifies the conjecture for planar curves.

Theorem 6 Let f ∈ C^{(3)}(I_{0}), where I_{0} is an interval and C_{f} := {(x, f(x)) : x ∈ I_{0}}. Let v > 1 and assume that dim {x ∈ I_{0} : f′′(x) = 0} ⩽ 2/(1 + v). Then

dim C_{f} ∩ S_{2}^{∗}(v) = \frac{2}{1 + v} .  (11)

Remark. Let ψ be an approximating function for which the limit

λ(ψ) := lim_{h→∞} \frac{− log ψ(h)}{log h}

exists and is positive. The quantity λ(ψ) is usually referred to as the order of 1/ψ and indicates the limiting behavior of the function 1/ψ at infinity. On making use of the fact that for any ε > 0,

h^{−λ(ψ)−ε} ≤ ψ(h) ≤ h^{−λ(ψ)+ε}  (12)

for all sufficiently large h, the above dimension results (Theorems 4 – 6) can be easily generalized to approximating functions ψ for which the order λ(ψ) exists. For example, Theorem 6 becomes

Theorem 6* 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 of order λ(ψ) > 1 and assume that dim {x ∈ I_{0} : f′′(x) = 0} ⩽ 2/(1 + λ(ψ)). Then

dim C_{f} ∩ S_{2}^{∗}(ψ) = \frac{2}{1 + λ(ψ)} .

Proof. Let v := λ(ψ) and fix ε > 0 such that v − ε > 1. In view of (12), it follows that

C_{f} ∩ S_{2}^{∗}(v + ε) ⊂ C_{f} ∩ S_{2}^{∗}(ψ) ⊂ C_{f} ∩ S_{2}^{∗}(v − ε) .

Theorem 6* now follows from these inclusions, (10) and (11), by letting ε → 0.

2 Preliminaries

First some useful notation. For any point r ∈ ℚ^{n} there exists a 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. As usual, C^{(n)}(I) will denote the set of n–times continuously differentiable functions defined on some interval I of ℝ. Also, as usual the Vinogradov symbols ≪ and ≫ will be used to indicate an inequality with an unspecified positive multiplicative constant. If a ≪ b and a ≫ b, we write a ≍ b and say that the quantities a and b are comparable.

2.1 Rational points close to a curve

The following estimates on the number of rational points close to a reasonably defined curve will be crucial towards establishing our convergence and (upper bound) dimension results.

Let I_{0} denote a finite, open interval of ℝ and let f be a function in C^{(3)}(I_{0}) such that

c_{1} < |f′′(x)| < c_{2} for all x ∈ I_{0} .  (13)

Here c_{1} and c_{2} are positive constants. Given an approximating function ψ and Q ∈ ℝ^{+} consider the counting function N_{f}(Q, ψ, I_{0}) given by

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

In short, the function N_{f}(Q, ψ, I_{0}) counts the number of rational points with bounded denominator lying within a specified neighbourhood of the curve C_{f} := {(x, f(x)) : x ∈ I_{0}} parameterized by f. Now let

lim_{t→+∞} ψ(t) = lim_{t→+∞} \frac{1}{tψ(t)} = 0.  (14)

In [8], Huxley obtains the following upper bound: For ε > 0 and Q sufficiently large

N_{f}(Q, ψ, I_{0}) ⩽ ψ(Q) Q^{2+ε}.  (15)

For this exact form of Huxley’s estimate we refer the reader to [3, §1.4]. In the case that the curve is the unit circle the above estimate can be sharpened. For n ∈ ℕ, let r(n) denote the number of representations of n as the sum of two squares. A simple consequence of Theorem A in [3, §A.1] is the following statement.

There is a constant C > 0 such that for any choice of real numbers Q and Ψ satisfying

Q^{−1} (log Q)^{260} ⩽ Ψ < 1  and  Q > 1 ,  (16)

one has that

∑_{Q<q⩽2Q} ∑_{\begin{matrix}n: \\ |q−\sqrt{n}|<Ψ \end{matrix}}r(n) ⩽ C Ψ Q^{2} .  (17)

Notice that if n is the sum of two square, say n = p_{1}^{2} +p_{2}^{2} then the inequality |q −\sqrt{n}| < Ψ appearing in (17) implies that the rational point (p_{1}/q, p_{2}/q) lies within a constant times Ψ/Q neighbourhood of the unit circle. Thus, we obtain the following sharpening of Huxley’s estimate.

If C_{f} is the unit circle and ψ(q) ⩾ q^{−1}(log q)^{260} for all sufficiently large q, then

N_{f}(Q, ψ, I_{0}) ≪ ψ(Q)Q^{2} .

In fact, on adapting the arguments of [3, §2] it is relatively straightforward to extend the statement to any planar curve C_{f} in Q; i.e to any non-degenerate rational quadric. However, we shall not make use of this stronger fact.

2.2 Ubiquitous systems

The divergence and (lower bound) dimension results stated in this paper will be established via a general technique developed in [3]. The ‘general technique’ is based on the notion of ‘ubiquity’ as introduced in [2].

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 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.

Throughout, ρ : ℝ^{+} → ℝ^{+} will denote a function satisfy 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 of radius r.

Definition 1 (Ubiquitous systems on the real line) Suppose there exists a ubiquitous 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.

In [3] the theory of ubiquity is developed 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 kth 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 above.

Definition 2 (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 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 satisfied for infinitely many α ∈ J. The following lemmas are stated and proved in [3, §3].

Lemma 1 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}. Suppose that (R_{C}(Φ), β) is a locally ubiquitous system with respect to ρ. 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_{C}(Φ), β, Ψ) | = |I_{0}|

whenever

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

Lemma 2 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}. Suppose that (R_{C}(Φ), β) is a locally ubiquitous system with respect to ρ and let Ψ be an approximating function. Then

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

Lemma 3 Let I_{0} denote a finite, open interval of ℝ and let f be a function in C^{(3)}(I_{0}) satisfying (13). Let ψ be an approximating function satisfying (14). Let C := {(x, f(x)) : x ∈ I_{0}}. With reference to the ubiquitous framework above, 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)) (18)

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. In Lemma 3, the curve C is obviously a planar curve. Also, given α = (p, q) ∈ J the associated resonant point R_{α} in the ubiquitous system is simply the rational point p/q in the plane. Furthermore, R := ℚ^{2}.

3 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 (13) with I_{0} replaced by I. However, without loss of generality and for clarity, we assume that f satisfies (13) on I_{0}.

We are given that ψ_{1} and ψ_{2} are approximating functions such that

∑_{h=1}^{∞} ψ_{1}(h)ψ_{2}(h) = ∞ .  (19)

Thus, at least one of the following two sums diverges:

∑_{h∈ℕ, ψ_{1}(h)⩾ψ_{2}(h)} ψ_{1}(h)ψ_{2}(h)  ∑_{h∈ℕ, ψ_{1}(h)⩽ψ_{2}(h)} ψ_{1}(h)ψ_{2}(h) .

Throughout, let us assume that the sum on the right is divergent. The argument below can easily be modified to deal with the case that only the sum on the left is divergent.

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

ψ_{2}(h) ⩾ ψ_{1}(h)  for all h ∈ ℕ .  (20)

Define the auxiliary function ψ_{1}^{∗} : h → ψ_{1}^{∗}(h) := min{ψ_{1}(h), ψ_{2}(h)}. Then the sum

∑_{h=1}^{∞} ψ_{1}^{∗}(h)ψ_{2}(h)

diverges since by assumption it contains a divergent sub-sum. It is readily verified that ψ_{1}^{∗} is an approximating function and that S_{2}(ψ_{1}^{∗}, ψ_{2}) ⊂ S_{2}(ψ_{1}, ψ_{2}). Thus to complete the proof of Theorem 1 it suffices to prove the result with ψ_{1} replaced by ψ_{1}^{∗}. Hence, without loss of generality, (20) can be assumed.

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

ψ_{i}(h) → 0  as  h → ∞  (i = 1, 2) .  (21)

Define the increasing function v : ℝ^{+} → ℝ^{+} as follows

v(h) := ∑_{t=1}^{[h]} ψ_{1}(t)ψ_{2}(t) .

In view of (19), lim_{t→∞} v(t) = ∞. Fix k ∈ ℕ. Then

∑_{t=k}^{m} \frac{ψ_{1}(t)ψ_{2}(t)}{v(t)} ⩾ ∑_{t=k}^{m} \frac{ψ_{1}(t)ψ_{2}(t)}{v(m)} = \frac{v(m) − v(k − 1)}{v(m)} → 1 as m → ∞.

Hence

∑_{t=k}^{∞} \frac{ψ_{1}(t)ψ_{2}(t)}{v(t)} ⩾ 1 for all k.

This implies that the sum ∑_{t=1}^{∞} ψ_{1}(t)ψ_{2}(t)/v(t) diverges. Next, for i = 1, 2 consider the functions

ψ_{i}^{∗} : h → ψ_{i}^{∗}(h) := ψ_{i}(h)/\sqrt{v(h)} .

Then both ψ_{1}^{∗}(h) and ψ_{2}^{∗}(h) are decreasing, tend to 0 as h → ∞ and ∑_{q=1}^{∞} ψ_{1}^{∗}(q)ψ_{2}^{∗}(q) = ∞. Furthermore S_{2}(ψ_{1}^{∗}, ψ_{2}^{∗}) ⊂ S_{2}(ψ_{1}, ψ_{2}). Therefore, it suffices to establish Theorem 1 for ψ_{1}^{∗}, ψ_{2}^{∗}.

Step 3. We show that there is no loss of generality in assuming that

ψ_{2}(h) ⩾ h^{−2/3} for all h.  (22)

To this end, define ψ̂_{2}(h) = max{ψ_{2}(h), h^{−2/3}}. In view of (20), it is readily verified that

S_{2}(ψ_{1}, ψ̂_{2}) ⊆ S_{2}(ψ_{1}, ψ_{2}) ∪ S_{2}(h ↦ h^{−2/3}, h ↦ h^{−2/3}) .

By Schmidt’s theorem [12], for almost all x ∈ I_{0} we have that

(x, f(x)) ∉ S_{2}(h ↦ h^{−2/3}, h ↦ h^{−2/3}) .

Hence

|{x ∈ I_{0} : (x, f(x)) ∈ S_{2}(ψ̂_{1}, ψ_{2})}| ≤ |{x ∈ I_{0} : (x, f(x)) ∈ S_{2}(ψ_{1}, ψ_{2})}| ,

and to complete the proof of Theorem 1 it suffices to prove that the set on the left has full measure. In turn, this justifies (22).

Step 4. In view of Steps 2 and 3 above, the function ψ_{2} satisfies (14) and Lemma 3 is applicable with ψ = ψ_{2}. By (19) and the fact that ψ_{1} and ψ_{2} are decreasing we obtain that

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

Hence

∑_{t=0}^{∞} 2^{t} ψ_{1}(2^{t})ψ_{2}(2^{t}) = ∞ .  (23)

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

u(h) = ∑_{t=0}^{[h]} 2^{t} ψ_{1}(2^{t})ψ_{2}(2^{t}) .

Trivially, lim_{t→∞} u(t) = ∞. On using the same argument as in Step 2 above we verify that

∑_{t=0}^{∞} \frac{2^{t} ψ_{1}(2^{t})ψ_{2}(2^{t})}{u(t)} = ∞ .  (24)

Now let Φ(t) := ψ_{2}(t)/t and ρ(t) := u(log_{2} t)/(t^{2}ψ_{2}(t)). By Lemma 3, (ℚ_{C}^{2}(Φ), β) is locally ubiquitous relative to ρ, where β is given by (18). Let Ψ(t) = ψ_{1}(t)/t. In view of (24),

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

Since ψ_{1} is decreasing,

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

Thus the conditions of Lemma 1 are satisfied and it follows that Λ(ℚ_{C}^{2}(Φ), β, Ψ) is of full measure. By definition and (20), the set Λ(ℚ_{C}^{2}(Φ), β, Ψ) consists of points x ∈ I_{0} such that the system

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

has infinitely many solutions p/q ∈ ℚ^{2}. Obviously for x ∈ Λ(ℚ_{C}^{2}(Φ), β, Ψ) the point (x, f(x)) is in S_{2}(2ψ_{1}, 2ψ_{2}). To complete the proof of Theorem 1 we simply apply what has already been proved to the approximating functions \frac{1}{2}ψ_{1} and \frac{1}{2}ψ_{2}.

♠

4 Proof of Theorem 3

By definition any rational quadric C ∈ Q is the image of either the unit circle C_{1} := {(x_{1}, x_{2}) ∈ ℝ^{2} : x_{1}^{2} + x_{2}^{2} = 1}, the parabola {(x_{1}, x_{2}) ∈ ℝ^{2} : x_{2} = x_{1}^{2}} or the hyperbola {(x_{1}, x_{2}) ∈ ℝ^{2} : x_{1}^{2} − x_{2}^{2} = 1} under a rational affine transformation of the plane. It is easily verified that the measure ‘zero’ statement of Theorem 3 is invariant under rational affine transformations of the plane. In view of this, it suffices to establish the statement of the theorem for the unit circle, the parabola and the hyperbola. Below, we only consider the case of the unit circle C_{1} and leave the hyperbola and parabola to the reader. The required modifications are relatively straightforward once the reader is armed with the arguments appearing in [3, §2.1].

From this point onwards C = C_{1} – the unit circle. First, notice that it suffices to prove the theorem for every arc of C_{1} given by C_{1}^{ε} = {(x, y) ∈ C_{1} : ε < x, y < 1 − ε} with ε > 0. Next, we are given that

∑_{q=1}^{∞} ψ(q) log q < ∞ .  (25)

Therefore, without loss of generality we can assume that

q^{−1}(log q)^{−3} < ψ(q) < q^{−1}(log q)^{−1} for sufficiently large q .  (26)

To see this, note that since ψ is decreasing

∑_{h/2<q<h} ψ(q) log q ⩾ ∑_{h/2<q<h} ψ(h) log(h/2) ≍ h ψ(h) log h

for every natural number h. In view of (25), we have that the left hand side of the above inequality tends to zero as h → ∞. It follows that

h ψ(h) log h → 0 as h → ∞ .

This establishes the right hand side inequality of (26). Next, if the left hand side inequality of (26) is not satisfied then we replace ψ with the auxiliary function

ψ̃: q → ψ̃(q) := max{ψ(q), q^{−1}(log q)^{−3}} .

Then ψ̃ is clearly an approximating function for which both (25) and the left hand side inequality of (26) are satisfied with ψ replaced by ψ̃. Furthermore,

S_{2}^{∗}(ψ̃) ⊃ S_{2}^{∗}(ψ) .

Thus it suffices to prove the theorem with ψ replaced by ψ̃. Hence, without loss of generality,

(26) can be assumed.

The lim sup set C_{1}^{ε} ∩ S_{2}^{∗}(ψ) has the following natural representation:

C_{1}^{ε} ∩ S_{2}^{∗}(ψ) = ⋂_{n=1}^{∞} ⋃_{q=n}^{∞} ⋃_{(p_{1},p_{2})∈ ℤ^{2}} \{(x, y) ∈ C_{1}^{ε} : |x − \frac{p_{1}}{q} | · |y − \frac{p_{2}}{q} | < \frac{ψ(q)}{q^{2}}\}.

Using the fact that ψ is decreasing, we have that for any n

C_{1}^{ε} ∩ S_{2}^{∗}(ψ) ⊂ ⋃_{t=n}^{∞} ⋃_{2^{t}⩽q<2^{t+1}} ⋃_{(p_{1},p_{2})∈ ℤ^{2}} \{(x, y) ∈ C_{1}^{ε} : |x − \frac{p_{1}}{q} | · |y − \frac{p_{2}}{q} | < \frac{ψ(2^{t})}{(2^{t})^{2}}\}. (27)

If t ∈ ℕ, (x, y) ∈ C_{1}^{ε}, q ∈ ℕ with 2^{t} ⩽ q < 2^{t+1} and |x − \frac{p_{1}}{q} | · |y − \frac{p_{2}}{q} | < \frac{ψ(2^{t})}{(2^{t})^{2}} for some (p_{1}, p_{2}) ∈ ℤ^{2}, then there is a unique integer m such that

2^{m−1} \frac{\sqrt{2 ψ(2^{t})}}{2^{t}} ⩽ |x − \frac{p_{1}}{q} | < 2^{m} \frac{\sqrt{2 ψ(2^{t})}}{2^{t}}.

Therefore for this number m we also have that

|y − \frac{p_{2}}{q} | < 2^{−m} \frac{\sqrt{2 ψ(2^{t})}}{2^{t}} .

The upshot of this is that

C_{1}^{ε} ∩ S_{2}^{∗}(ψ) ⊂ ⋃_{t=n}^{∞} ⋃_{2^{t}⩽q<2^{t+1}} ⋃_{(p_{1},p_{2})∈ ℤ^{2}} ⋃_{m=−∞}^{+∞} C_{1}^{ε} ∩ S(q, p_{1}, p_{2}, m) ,  (28)

where

S(q, p_{1}, p_{2}, m) = \{(x, y) ∈ ℝ^{2} : |x − \frac{p_{1}}{q} | < 2^{m} \frac{\sqrt{2 ψ(2^{t})}}{2^{t}}, |y − \frac{p_{2}}{q} | < 2^{−m} \frac{\sqrt{2 ψ(2^{t})}}{2^{t}}\}

and t is uniquely defined by 2^{t} ⩽ q < 2^{t+1}. The aim is to show that the Lebesgue measure | . |_{C_{1}^{ε}} of the R.H.S. of (28) tends to zero as n → ∞. Since for each n the R.H.S. of (28) is a cover for C_{1}^{ε} ∩ S_{2}^{∗}(ψ), it follows that |C_{1}^{ε} ∩ S_{2}^{∗}(ψ)|_{C_{1}^{ε}} = 0 as required. To proceed, we consider two cases. Namely, case (a): m ∈ ℤ such that

2^{−|m|} ⩾ t\sqrt{ψ(2^{t})}  (29)

and case (b): m ∈ ℤ such that

2^{−|m|} ⩽ t\sqrt{ψ(2^{t})} .  (30)

Case (a) : First, observe that (29) together with (26) implies that

t ≥ 2 |m| .  (31)

Next, it is a simple mater to see that

| C_{1}^{ε} ∩ S(q, p_{1}, p_{2}, m)|_{C_{1}^{ε}} ≪ 2^{−|m|} \frac{\sqrt{ψ(2^{t})}}{2^{t}} .  (32)

The implied constant depends on only ε and is therefore irrelevant to the rest of the argument.

Given t and m, let N(t, m) denote the number of triples (q, p_{1}, p_{2}) with 2^{t} ⩽ q < 2^{t+1} such that C_{1}^{ε} ∩ S(q, p_{1}, p_{2}, m) ≠ ∅. Suppose that C_{1}^{ε} ∩ S(q, p_{1}, p_{2}, m) ≠ ∅. Then for some (x, y) ∈ C_{1}^{ε} and θ_{1}, θ_{2} satisfying −1 < θ_{1}, θ_{2} < 1, we have that

x = \frac{p_{1}}{q} + θ_{1} 2^{|m|} \frac{\sqrt{2 ψ(2^{t})}}{2^{t}} , y = \frac{p_{1}}{q} + θ_{2} 2^{|m|} \frac{\sqrt{2 ψ(2^{t})}}{2^{t}} .

Hence

1 = x^{2} + y^{2}
= ∑_{i=1}^{2} (\frac{p_{i}}{q} + θ_{i} 2^{|m|} \frac{\sqrt{2 ψ(2^{t})}}{2^{t}} )^{2}

= \frac{p_{1}^{2} + p_{2}^{2}}{q^{2}} + \frac{p_{1}θ_{1} + p_{2}θ_{2}}{q} 2^{|m|+1} \frac{\sqrt{2 ψ(2^{t})}}{2^{t}} + (θ_{1}^{2} + θ_{2}^{2}) 2^{2|m|+1} \frac{ψ(2^{t})}{2^{2t}} .

It follows that

| q^{2} − p_{1}^{2} + p_{2}^{2} | ≪ q max\{|p_{1}|, |p_{2}|\} 2^{|m|−t} \sqrt{ψ(2^{t})} + q^{2} 2^{2|m|−2t} ψ(2^{t}) .

On dividing both sides of the inequality by q + \sqrt{p_{1}^{2} + p_{2}^{2}} and using the fact that \sqrt{p_{1}^{2} + p_{2}^{2}} ⩾ max{|p_{1}|, |p_{2}|} we obtain that

| q − \sqrt{p_{1}^{2} + p_{2}^{2}} | ≪ 2^{t} U_{m,t} (1 + U_{m,t}) ,

where

U_{m,t} := 2^{−t} 2^{|m|} \sqrt{ψ(2^{t})} ≤^{(29)} 2^{−t} t^{−1} < 1 .

Thus

| q − \sqrt{p_{1}^{2} + p_{2}^{2}} | ≪ 2^{|m|} \sqrt{ψ(2^{t})} ≤^{(29)} t^{−1} .  (33)

The upshot of this is that there exists an absolute constant c > 0 such that

N(t, m) ≪ ∑_{2^{t}⩽q<2^{t+1}} ∑_{|q−\sqrt{n}|<c2^{|m|}\sqrt{ψ(2^{t})}} r(n) .

Now set Q = 2^{t} and Ψ = c2^{|m|}\sqrt{ψ(2^{t})}. In view of (26) and (33) we have that (16) is satisfied for all sufficiently large t, independently of m. Hence, (17) implies that

N(t, m) ≪ 2^{|m|} 2^{2t} \sqrt{ψ(2^{t})} ,  (34)

where the implied constant is independent of both t and m.

It now follows, via (32) and (34) that the Lebesgue measure | . |_{C_{1}^{ε}} of the R.H.S. of (28) restricted to case (a) is bounded above by

∑_{t=n}^{∞} ∑_{m∈Case(a)}2^{−|m|} \frac{\sqrt{ψ(2^{t})}}{2^{t}} × 2^{|m|} 2^{2t}\sqrt{ψ(2^{t})} ≪ ∑_{t=n}^{∞} ∑_{m∈Case(a)}2^{t} ψ(2^{t})

≪^{(31)} ∑_{t=n}^{∞} t 2^{t}ψ(2^{t})

≍ ∑_{q=2^{n}}^{∞} ψ(q) log q .

The above comparability follows from the fact that ψ is an approximating function and therefore decreasing. In view of (25)

∑_{q=2^{n}}^{∞} ψ(q) log q → 0  as  n → ∞ ,

and so the Lebesgue measure | . |_{C_{1}^{ε}} of the R.H.S. of (28) restricted to case (a) tends to zero as n → ∞.

Case (b) : In view of (30), we have that

S(q, p_{1}, p_{2}, m) ⊂ S′(q, p_{1}) × [0, 1] ∪ [0, 1] × S′(q, p_{2})  (35)

where

S′(q, p) = \{s ∈ [0, 1] : |s − \frac{p}{q}| < \frac{2 t ψ(2^{t})}{2^{t}}\}  (36)

Thus, the R.H.S. of (28) restricted to case (b) is contained in the following set:

⋃_{t=n}^{∞} ⋃_{2^{t}⩽q<2^{t+1}} ⋃_{(p_{1},p_{2})∈ ℤ^{2}} C_{1}^{ε} ∩ ( S′(q, p_{1}) × [0, 1] ∪ [0, 1] × S′(q, p_{2}) ) .  (37)

It is readily verified that for any choice of p_{1}, p_{2} and q appearing in (37),

| C_{1}^{ε} ∩ ( S′(q, p_{1}) × [0, 1] ∪ [0, 1] × S′(q, p_{2}) ) |_{C_{1}^{ε}} ≪ t ψ(2^{t})/2^{t} .

The implied constant depends only on ε and is therefore irrelevant. Furthermore, for a fixed t and q in (37) the number of (p_{1}, p_{2}) ∈ ℤ^{2} for which the sets

C_{1}^{ε} ∩ ( S′(q, p_{1}) × [0, 1] ∪ [0, 1] × S′(q, p_{2}) )

are non-empty and disjoint is ≪ q. It now follows that the Lebesgue measure | . |_{C_{1}^{ε}} of the set given by (37) is bounded above by

∑_{t=n}^{∞} t ψ(2^{t}) ≍ ∑_{q=2^{n}}^{∞} ψ(q) log q → 0 as n → ∞ .

Hence the Lebesgue measure | . |_{C_{1}^{ε}} of the R.H.S. of (28) restricted to case (b) tends to zero as n → ∞.

The upshot of cases (a) and (b) is that the Lebesgue measure | . |_{C_{1}^{ε}} of the R.H.S. of (28) tends to zero as n → ∞ and so

|C_{1}^{ε} ∩ S_{2}^{∗}(ψ)|_{C_{1}^{ε}} = 0 .

This completes the proof of Theorem 3.

♠

5 Proof of Theorem 2

The divergence part of Theorem 2 is a consequence of Theorem 1. Thus we proceed with establishing the convergence part of Theorem 2. To a certain degree the proof of this follows the same line of argument as the proof of Theorem 3. In particular, it suffices to establish the convergent statement of the theorem for the unit circle, the parabola and the hyperbola. As in the proof of Theorem 3, we consider the case of the unit circle C_{1} only and leave the hyperbola and parabola to the reader.

Let C = C_{1} – the unit circle, and notice that it suffices to prove the theorem for every arc of C_{1} given by C_{1}^{ε} = {(x, y) ∈ C_{1} : ε < x, y < 1 − ε} with ε > 0. For the sake of convenience, let ψ := ψ_{1} and ϕ := ψ_{2}. It is clear that

S_{2}(ψ, ϕ) ⊂ S_{2}(ψ^{∗}, ψ_{∗}) ∪ S_{2}(ψ_{∗}, ψ^{∗})

where

ψ_{∗} = min{ψ, ϕ} , ψ^{∗} = max{ψ, ϕ}.

Since ψ^{∗}ψ_{∗} = ψϕ, we have that ∑ ψ^{∗}(q)ψ_{∗}(q) < ∞. Thus to prove the theorem it suffices to prove that both the sets C_{1}^{ε} ∩ S_{2}(ψ^{∗}, ψ_{∗}) and C_{1}^{ε} ∩ S_{2}(ψ_{∗}, ψ^{∗}) are of Lebesgue measure | . |_{C_{1}^{ε}} zero. We will consider one of these two sets – the other case is similar. Thus, without loss of generality we assume that

ψ(q) ⩾ ϕ(q) for all q .

Since ∑_{q=1}^{∞} ψ(q) ϕ(q) < ∞ and both ψ and ϕ are decreasing we have that ψ(q)ϕ(q) < q^{−1} for all sufficiently large q. Hence

ϕ(q) ⩽ q^{−1/2} for sufficiently large q .

Further, we can assume that

ψ(q) ⩾ q^{−2/3}  for all q ∈ ℕ .  (38)

To see this, consider the auxiliary function ψ̃ : q → ψ̃(q) := max{ψ(q), q^{−2/3}}. Clearly, ψ̃ is an approximating function, satisfies (38) and

S_{2}(ψ, ϕ) ⊂ S_{2}(ψ̃, ϕ) .

Moreover,

\begin{matrix}∑_{q=1}^{∞} ψ̃(q) ϕ(q) ⩽ & ∑_{q=1}^{∞} ψ(q)ϕ(q) + ∑_{q=1}^{∞} q^{−2/3}ϕ(q) \\ ≪ & ∑_{q=1}^{∞} ψ(q)ϕ(q) + ∑_{q=1}^{∞} q^{−2/3}q^{−1/2} < ∞ . \end{matrix}

Thus, it suffices to prove the convergence part of Theorem 2 with ψ := ψ_{1} replaced ψ̃. Hence, without loss of generality, (38) can be assumed.

In analogy to (27), it is readily verified that for any n ≥ 1

C_{1}^{ε} ∩ S_{2}(ψ, ϕ) ⊂ ⋃_{t=n}^{∞} ⋃_{2^{t}⩽q<2^{t+1}} ⋃_{(p_{1},p_{2})∈ ℤ^{2}} C_{1}^{ε} ∩ S_{2}(p_{1}, p_{2}, q) ,  (39)

where

S_{2}(p_{1}, p_{2}, q) = \{(x, y) ∈ ℝ^{2} : |x − \frac{p_{1}}{q} | < \frac{ψ(2^{t})}{2^{t}}, |y − \frac{p_{2}}{q} | < \frac{ϕ(2^{t})}{2^{t}}\}

and t is uniquely defined by 2^{t} ⩽ q < 2^{t+1}. Next, in analogy to (32), we verify that

| C_{1}^{ε} ∩ S_{2}(q, p_{1}, p_{2})|_{C_{1}^{ε}} ≪ \frac{ϕ(2^{t})}{2^{t}} .  (40)

Again, the implied constant depends only on ε and is therefore irrelevant to the rest of the argument. For t fixed , let N(t) denote the number of triples (q, p_{1}, p_{2}) with 2^{t} ⩽ q < 2^{t+1} such that C_{1}^{ε} ∩ S(q, p_{1}, p_{2}) ≠ ∅. On modifying the argument used to establish (33) and (34) in the proof of Theorem 3, one obtains that

N(t) ≪ 2^{2t}ψ(2^{t}) .  (41)

It is worth stressing that the argument within the proof of Theorem 3 is much simplified in the current situation due to the absence of the additional parameter m.

The upshot of the above inclusions and estimates is that

| C_{1}^{ε} ∩ S_{2}(ψ, ϕ)|_{C_{1}^{ε}}  ≪^{(39)} ∑_{t=n}^{∞} ∑_{2^{t}⩽q<2^{t+1}} ⋃_{(p_{1},p_{2})∈ ℤ^{2}} |C_{1}^{ε} ∩ S_{2}(q, p_{1}, p_{2})|

≪^{(40)} ∑_{t=n}^{∞} N(t) \frac{ϕ(2^{t})}{2^{t}}

≪^{(41)} ∑_{t=n}^{∞} 2^{t}ψ(2^{t})ϕ(2^{t})

≍ ∑_{q=2^{n}}^{∞} ψ(q)ϕ(q)

Since ∑_{q=2^{n}}^{∞} ψ(q)ϕ(q) < ∞, we have that ∑_{q=2^{n}}^{∞} ψ(q)ϕ(q) → 0 as n → ∞. Thus,

| C_{1}^{ε} ∩ S_{2}(ψ, ϕ)|_{C_{1}^{ε}} = 0 .

This completes the proof of the theorem.

♠

6 Proofs of Theorems 4 – 6

6.1 Proof of Theorem 4

The statement of the theorem will follow on establishing the upper and lower bounds for the dimension separately. Without loss of generality we can assume that f satisfies (13) on I_{0} (see [3, §5] if necessary) and that v_{1} ⩾ v_{2}. In view of the latter, our aim is to show that

dim C_{f} ∩ S_{2}(v_{1}, v_{2}) = \frac{2 − v_{2}}{1 + v_{1}} .

The upper bound. For a point p/q ∈ ℚ^{2}, define

σ(p/q) := {(x, y) ∈ ℝ^{2} : |x − p_{1}/q| < q^{−v_{1}−1}, |y − p_{2}/q| < q^{−v_{2}−1}}.

In view of (13) and the fact that I_{0} is a bounded interval we have that f′ is bounded on I_{0} and so |C_{f} ∩ σ(p/q)| ≪ q^{−v_{1}−1}. Clearly, if σ(p/q) ∩ C_{f} ≠ ∅ then the distance of p/q from C_{f} is at most a constant times q^{−1−v_{2}}. Let ε > 0. In view of (15) we have that for t sufficiently large the number p/q ∈ ℚ^{2} with 2^{t} ⩽ q < 2^{t+1} and σ(p/q) ∩ C_{f} ≠ ∅ is at most 2^{t(2+ε−v_{2})}. Now let

η := \frac{2 − v_{2} + 2ε}{1 + v_{1}} .

Then

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

≪ ∑_{t=0}^{∞} 2^{t(2+ε−v_{2})} · 2^{t(−1−v_{1})η} = ∑_{t=0}^{∞} 2^{−tε} < ∞ .

By the Hausdorff–Cantelli Lemma [4, p. 68], dim C_{f} ∩ S_{2}(v_{1}, v_{2}) ⩽ η. As ε > 0 is arbitrary,

dim C_{f} ∩ S_{2}(v_{1}, v_{2}) ⩽ \frac{2 − v_{2}}{1 + v_{1}} .  (42)

The lower bound. Firstly, with reference to Lemma 3 let ψ(t) := \frac{1}{2}t^{−v_{2}} and u(t) := t^{ε} where ε > 0 is arbitrary. Thus Φ(t) = \frac{1}{2}t^{−1−v_{2}} and ρ(t) := \frac{1}{2}t^{−2+v_{2}+ε}. Since 0 < v_{2} = min(v_{1}, v_{2}) < 1, the approximating function ψ satisfies (14) and it follows that (ℚ_{C}^{2}(Φ), β) is locally ubiquitous with respect to ρ. Next, let Φ(t) := \frac{1}{2}t^{−1−v_{1}}. Then Lemma 2 implies that

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

As ε > 0 can be made arbitrarily small, we have that dim Λ (R_{C}(Φ), β, Ψ) ⩾ \frac{2−v_{2}}{1+v_{1}} . Finally, it is readily verified that

Λ̃ (R_{C}(Φ), β, Ψ) := {(x, f(x)) : x ∈ Λ (R_{C}(Φ), β, Ψ)} ⊂ C_{f} ∩ S_{2}(v_{1}, v_{2}).

Hence

dim C_{f} ∩ S_{2}(v_{1}, v_{2}) ≥ dim Λ̃ (R_{C}(Φ), β, Ψ) = dim Λ (R_{C}(Φ), β, Ψ) ≥ \frac{2 − v_{2}}{1 + v_{1}} .

The equality here is justified by the fact that the map x ↦ (x, f(x)) is locally bi-Lipshitz.

♠

6.2 Proof of Theorem 5

Let m = dim M. Since M is a Lipshitz manifold in ℝ^{n}, there exists a local parameterization of M of the form f = (x_{1}, …, x_{m}, f_{m+1}, …, f_{n}) where f is an invertible continuous map of x_{1}, …, x_{m} defined on ℝ^{m} such that f^{−1} satisfies the Lipshitz condition. It is easy to verify that any point on M with x_{1} ∈ S_{1}^{∗}(v) belongs to S_{n}^{∗}(v). Therefore,

B := f(S_{1}^{∗}(v) × ℝ^{m−1}) ⊂ S_{n}^{∗}(v) ∩ M .

Since f^{−1} is a Lipshitz map and f^{−1}(B) = S_{1}^{∗}(v) × ℝ^{m−1}, it follows that for v ≥ 1

dim S_{n}^{∗}(v) ∩ M ⩾ dim B ⩾ dim S_{1}^{∗}(v) × ℝ^{m−1} = m − 1 + dim S_{1}^{∗}(v)  (43)
= dim M − 1 + \frac{2}{1 + v} .

The fact that dim S_{1}^{∗}(v) = 2/(1 + v) is the Jarník–Besicovitch theorem (see §1.1).

♠

6.3 Proof of Theorem 6

The lower bound is a trivial consequence of Theorem 5. Alternatively, it follows from Theorem 4 with v_{1} = ε, v_{2} = v − ε and then letting ε → 0.

To establish the complementary upper bound we fix v > 1 and without loss of generality assume that f satisfies (13) on I_{0}. The case that v = 1 is trivial. Now fix ε such that

0 < ε < min{1/(1 + v), 1/5}  and  v − ε > 1 .

The following inclusions readily follow from the definitions of S_{2}^{∗}(v) and S_{2}(v_{1}, v_{2}) :

S_{2}^{∗}(v) ⊂ S_{2}(v − ε, 0) ∪ S_{2}(0, v − ε) ∪ ⋃_{t=−t_{0}}^{t_{0}} S_{2}(v_{1}(t), v_{2}(t)) ⊂ S_{2}^{∗}(v − ε) , (44)

where t_{0} > 0 is the unique positive integer satisfying \frac{v}{2ε} − \frac{3}{2} ≤ t_{0} < \frac{v}{2ε} − \frac{1}{2} and

v_{1}(t) := \frac{v}{2} − \frac{(2t+1)ε}{2} v_{2}(t) := \frac{v}{2} + \frac{(2t−1)ε}{2} .

The required upper bound will follow on establishing the corresponding upper bounds for the sets ‘between the inclusions’ of (44). For this we will repeatedly apply Theorem 4.

First, consider the sets S_{2}(v_{1}(t), v_{2}(t)) for t ⩾ 0 (the case t < 0 is similar). So, v_{1}(t) ≤ v_{2}(t). Assume for the moment that v_{1}(t) < 1. Then v_{1}(t) ∈ (0, 1) and since v_{1}(t) + v_{2}(t) = v − ε > 1 we have via Theorem 4 that

dim S_{2}(v_{1}(t), v_{2}(t)) ∩ C_{f} = \frac{2 − v_{1}(t)}{1 + v_{2}(t)} = \frac{2 − v_{1}(t)}{1 + v − ε − v_{1}(t)} ≤ \frac{2}{1 + v − ε} .

Now suppose that v_{1}(t) ≥ 1. It follows from the definition of v_{1}(t) that v > 2. Trivially, S_{2}(v_{1}(t), v_{2}(t)) ⊂ S_{2}(1 − ε, v/2). Now, v/2 > 1 − ε > 0 and on applying Theorem 4 we have that

dim S_{2}(v_{1}(t), v_{2}(t)) ∩ C_{f} ≤ dim S_{2}(1 − ε, v/2) ∩ C_{f} = \frac{2 + 2ε}{2 + v} ⩽ \frac{2}{1 + v} .

Next, we consider the set S_{2}(v − ε, 0) – the case of S_{2}(0, v − ε) is similar. By definition, S_{2}(v − ε, 0) = S_{1}(v − ε) × ℝ and so

dim S_{2}(v − ε, 0) ∩ C_{f} ⩽ dim S_{1}(v − ε) = \frac{2}{1 + v − ε} .

The upshot is that

dim S_{2}^{∗}(v) ∩ C_{f} ≤ max \{\frac{2}{1 + v − ε}, \frac{2}{1 + v} \} = \frac{2}{1 + v − ε} ,

and since ε can be made arbitrarily small the required upper bound follows.

♠

7 Final remarks: the dual form of approximation

In view of Khintchine’s transference principle [13], Theorem KM can be reformulated for the dual form of approximation:

Theorem KM′ Let M be a non-degenerate manifold in ℝ^{n}. Then for any v > 1 for almost every point (y_{1}, …, y_{n}) ∈ M the inequality

∥a_{1}y_{1} + … + a_{n}y_{n}∥ < Π_{+}(a)^{−v}  (45)

has only finite number of solutions a = (a_{1}, …, a_{n}) ∈ ℤ^{n}, where

Π_{+}(a) := ∏_{i=1}^{n} max{1, |a_{i}|} .

The problems S1 and S2 considered in §1.2 above can therefore be reformulated for the dual form of approximation. Given an approximating function ψ, consider the inequality

∥a_{1}y_{1} + … + a_{n}y_{n}∥ < ψ(Π_{+}(a)) .  (46)

Let

L_{n}^{∗}(ψ) := {y ∈ ℝ^{n} : (46) holds for infinitely many a ∈ ℤ^{n}}

and

L_{n}^{∗}(v) := S^{∗}(q ↦ q^{−v}) := {y ∈ ℝ^{n} : (45) holds for infinitely many a ∈ ℤ^{n}} .

Problem D1 : Given a non-degenerate manifold M ⊂ ℝ^{n} and v > 1, what is the Hausdorff dimension of L^{∗}(v) ∩ M ?

Note that above theorem of Kleinbock and Margulis only implies that the Lebesgue measure of L^{∗}(v) ∩ M is zero.

Problem D2 : Given a non-degenerate manifold M ⊂ ℝ^{n} and an approximating function ψ, what is the weakest condition under which L^{∗}(ψ) ∩ M is of Lebesgue measure zero ?

Regarding Problem D1 the following general lower bound can be established:

Theorem 7 Let M be arbitrary manifold in ℝ^{n}. Then for any v > 1

dim M ∩ L_{n}^{∗}(v) ⩾ dim M − 1 + \frac{2}{1 + v} .  (47)

The proof of Theorem 7 follows the same line of reasoning as that of Theorem 5 and is left to the reader. It is highly likely that the inequality given by (47) is in fact an equality. For n = 2, that this is indeed the case is easily verified by modifying the arguments of [14]. However, the general case (n ≥ 3) seems to be a difficult problem.

Regarding Problem D2 a general Khintchine-Groshev type theorem for convergence has been established in [5]. This states that in Theorem KM′ above one can replace (45) with (46) whenever

∑_{h=1}^{∞} ψ(h) log^{n−1} h < ∞ .

The divergence counterpart remains an open problem even for planar curves.

Acknowledgements. SV would like to thank Ayesha (Dorothy) and Iona (Tinman) for allowing him into their wonderful world of Oz and for constantly restuffing his straw. Also, he’d like to thank Bridget (Wicked Witch) for her support and friendship. Finally, many thanks to Geraldine and Peter for their generosity and introducing us creatures from Oz to Rasquera.

References

[1] R.C. Baker : Dirichlet’s theorem on Diophantine approximation, Math. Proc. Cam. Phil. Soc. 83 (1978), 37–59.

[2] V. Beresnevich, H. Dickinson, and S. Velani : Measure Theoretic Laws for limsup sets, Pre-print (97pp): arkiv:math.NT/0401118. To appear: Memoirs of the AMS.

[3] V. Beresnevich, H. Dickinson, and S. Velani : Diophantine approximation on planar curves and the distribution of rational points, Pre-print (52pp): arkiv:math.NT/0401148.

[4] V.I. Bernik and M.M. Dodson : Metric Diophantine approximation on manifolds, Cambridge Tracts in Mathematics, vol. 137, Cambridge University Press, Cambridge, 1999.

[5] V.I. Bernik, D.Y. Kleinbock, and G.A. Margulis : Khintchine–type theorems on manifolds: the convergence case for standard and multiplicative versions, International Mathematics Research Notices (2001), no. 9, 453–486.

[6] J.D. Bovey and M.M. Dodson : The fractional dimension of sets whose simultaneous rational approximations have errors with a small product, Bull. London Math. Soc. 10 (1978), no. 2, 213–218.

[7] P.X. Gallagher : Metric simultaneous Diophantine approximation, Jour. London Math. Soc. 37 (1962), 387–390.

[8] M.N. Huxley : Area. Lattice points and exponential sums, Oxford, 1996.

[9] D.Y. Kleinbock and G.A. Margulis : Flows on homogeneous spaces and Diophantine approximation on manifolds, Ann. Math. 148 (1998), 339–360.

[10] A. Khintchine : Zur metrischen Theorie der diophantischen Approximationen, Math. Z. 24 (1926), 706–714.

[11] B.P. Rynne : Hausdorff dimension and generalized simultaneous Diophantine approximation. Bull. London Math. Soc. 30 (1998), no. 4, 365–376.

[12] W.M. Schmidt : Metrische Sätze über simultane Approximation abhängiger Größen, Monatsch. Math. 63 (1964), 154–166.

[13] V.G. Sprindžuk : Metric theory of Diophantine approximation, John Wiley & Sons, New York-Toronto-London, 1979, (English transl.).

[14] Kunrui Yu : A note on a problem of Baker in metrical number theory, Math. Proc. Cambridge Philos. Soc. 90 (1981), 215–227.

Victor V. Beresnevich: Institute of Mathematics, Academy of Sciences of Belarus, 220072, Surganova 11, Minsk, Belarus.

e-mail: beresnevich@im.bas-net.by

Sanju L. Velani: Department of Mathematics, University of York, Heslington, York, YO10 5DD, England.

e-mail: slv3@york.ac.uk