Diophantine approximation and badly approximable sets

Simon Kristensen^{a,1} Rebecca Thorn^{b} Sanju Velani^{c,∗,2}

aSchool of Mathematics, The University of Edinburgh, James Clerk Maxwell Building, King’s Buildings, Mayfield Road, Edinburgh EH9 3JZ, UK

bSchool of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, UK

cDepartment of Mathematics, University of York, Heslington, York YO10 5DD, UK

Abstract

Let (X, d) be a metric space and (Ω, d) a compact subspace of X which supports a non-atomic finite measure m. We consider ‘natural’ classes of badly approximable subsets of Ω. Loosely speaking, these consist of points in Ω which ‘stay clear’ of some given set of points in X. The classical set Bad of ‘badly approximable’ numbers in the theory of Diophantine approximation falls within our framework as do the sets Bad(i, j) of simultaneously badly approximable numbers. Under various natural conditions we prove that the badly approximable subsets of Ω have full Hausdorff dimension. Applications of our general framework include those from number theory (classical, complex, p-adic and formal power series) and dynamical systems (iterated function schemes, rational maps and Kleinian groups).

Key words: Diophantine approximation, Hausdorff dimension, badly approximable elements, dynamical systems.

1991 MSC: 11J83, 11J61, 28A80, 37C45, 37F

Dedicated to Lalji and Manchaben on their 70 plus birthdays

∗ Corresponding author Email addresses: Simon.Kristensen@ed.ac.uk (Simon Kristensen),

r.e.thorn@qmul.ac.uk (Rebecca Thorn), slv3@york.ac.uk (Sanju Velani).

1 William Gordon Seggie Brown Fellow 2 Royal Society University Research Fellow

Preprint submitted to Elsevier Science 13 April 2005

1 Introduction

1.1 The setup and the problem

Let (X, d) be a metric space and (Ω, d) a compact subspace of X which contains the support of a non-atomic finite measure m. Let R = {R_{α} ∈ X : α ∈ J} be a family of subsets R_{α} of X indexed by an infinite, countable set J. The sets R_{α} will be referred to as resonant sets. Next, let β : J → ℝ^{+} : α → β_{α} be a positive function on J. To avoid pathological situations within our framework, we shall assume that the number of α ∈ J with β_{α} bounded above is finite – thus β_{α} tends to infinity as α runs through J. Given a real, positive function ρ : ℝ^{+} → ℝ^{+} : r → ρ(r) such that ρ(r) → 0 as r → ∞ and that ρ is decreasing for r large enough, consider the set

Bad^{∗}(R, β, ρ) := {x ∈ Ω : ∃ c(x) > 0 such that d(x, R_{α}) ≥ c(x)ρ(β_{α}) for all α ∈ J} ,

where d(x, R_{α}) := inf_{a∈R_{α}} d(x, a). Loosely speaking, in the case that the resonant sets are points, Bad^{∗}(R, β, ρ) consists of points in Ω which ‘stay clear’ of ‘ρ-balls’ centred at resonant points. Notice that since the number of α ∈ J with β_{α} bounded above is finite and ρ is eventually decreasing, the number of α ∈ J with ρ(β_{α}) ≥ ε > 0 is finite. In view of this, without loss of generality we shall assume that the sup_{α∈J} ρ(β_{α}) is finite. Otherwise, if ρ(β_{α}) can get arbitrarily large, then trivially Bad^{∗}(R, β, ρ) = ∅ – recall that Ω is compact and so is bounded.

The set Bad^{∗}(R, β, ρ) is easily seen to be a generalization of the classical set Bad of badly approximable numbers. Recall, a real number x is said to be badly approximable if there exists a constant c(x) > 0 such that |x − p/q| ≥ c(x)/q^{2} for all rational p/q. A result of Jarník [10] states that the Hausdorff dimension of Bad is maximal; i.e. dim Bad = 1. Our initial aim is to find a suitably general framework which allows us to conclude that dim Bad^{∗}(R, β, ρ) = dim Ω; that is to say that the set of badly approximable points in Ω is of maximal dimension. To a certain extent, this paper complements [3] in which a general framework for establishing measure theoretic laws for ‘well approximable’ sets is established.

A few words about our chosen notation are in order. In the above setup and its generalization in §2, the sets of badly approximable elements will be denoted by Bad^{∗} followed by the appropriate variables in brackets. In applications we define a set, usually denoted by Bad with appropriate arguments, and show that this set may be realized as a specialization of a general set Bad^{∗}.

1.2 The conditions on the setup

Throughout, a ball B(c, r) with centre c and radius r is defined to be the set {x ∈ X : d(c, x) ≤ r}. Thus all balls will be assumed to be closed unless stated otherwise and by definition a ball is a subset of X. The following conditions on the measure m and the function ρ will play a central role in our work.

(A) There exist strictly positive constants δ and r_{0} such that for c ∈ Ω and

r ≤ r_{0}

a r^{δ} ≤ m(B(c, r)) ≤ b r^{δ} ,

where 0 < a ≤ 1 ≤ b are constants independent of the ball.

It is easily verified that if the measure m supported on Ω is of type (A) then dim Ω = δ. Trivially, this implies that dim X ≥ δ. See §3 for the details.

(B) For k > 1 sufficiently large and any integer n ≥ 1,

λ^{l}(k) ≤ \frac{ρ(k^{n})}{ρ(k^{n+1})} ≤ λ^{u}(k)

where λ^{l} and λ^{u} are lower and upper bounds depending only on k such that λ^{l}(k) → ∞ as k → ∞.

Note that this condition on ρ is satisfied by any function satisfying the following ‘regularity’ condition. There exist a constant k > 1 such that for r sufficiently large

λ^{l} ≤ \frac{ρ(r)}{ρ(k r)} ≤ λ^{u} ,

where 1 < λ^{l} ≤ λ^{u} are constants independent of r but may depend on k.

1.3 The result

First some useful notation. For any k > 1 let B_{n} := {x ∈ Ω : d(c, x) ≤ ρ(k^{n})} denote a generic closed ball of radius ρ(k^{n}) with centre c in Ω and for θ ∈ ℝ^{+}, let θB_{n} := {x ∈ Ω : d(c, x) ≤ θρ(k^{n})} denote the ball B_{n} scaled by θ. Notice, that by definition any generic ball B_{n} is a subset of Ω. Also, for n ≥ 1 let J(n) := {α ∈ J : k^{n−1} ≤ β_{α} < k^{n}}.

Theorem 1 Let (X, d) be a metric space and (Ω, d, m) a compact measure subspace of X. Let the measure m and the function ρ satisfy conditions (A) and (B) respectively. For k ≥ k_{0} > 1, suppose there exists some θ ∈ ℝ^{+} so that for n ≥ 1 and any ball B_{n} there exists a collection C(θB_{n}) of disjoint balls 2θB_{n+1} contained within θB_{n} satisfying

#C(θB_{n}) ≥ κ_{1} (\frac{ρ(k^{n})}{ρ(k^{n+1})})^{δ}  (1)

and

# \{2θB_{n+1} ⊂ C(θB_{n}) : min_{α∈J(n+1)} d(c, R_{α}) ≤ 2θρ(k^{n+1})\} ≤ κ_{2} (\frac{ρ(k^{n})}{ρ(k^{n+1})})^{δ} ,

(2)

where 0 < κ_{2} < κ_{1} are absolute constants independent of k and n. Furthermore, suppose dim(∪_{α∈J}R_{α}) < δ. Then

dim Bad^{∗}(R, β, ρ) = δ .

Remarks:

(i) In applications, the ‘scaling factor’ θ is usually dependent on k – see the basic example below. For k sufficiently large, it is always possible to find the collection C(θB_{n}) satisfying condition (1) – see §3 for the details. Finally, note that in the case that the resonant sets are points dim(∪_{α∈J}R_{α}) = 0 and the hypothesis that dim(∪_{α∈J}R_{α}) < δ is trivially satisfied. This follows from the fact that the indexing set J is countable.

(ii) We suspect that Theorem 1 can be established using Schmidt games [20] – a standard mechanism in the subject to prove such full dimension results. However, we will deduce the result from a more general one (Theorem 2 below) which we have not been able to prove using Schmidt games. In fact, it is not at all clear that the Schmidt games mechanism is even applicable.

(iii) Here and in subsequent theorems, we consider families of general resonant sets R_{α}. However, in all the applications considered in §5, the resonant sets are assumed to be points. There are natural problems of the same type where this is not the case. For example, when considering the classical problem of approximation of systems of linear forms over ℝ the resonant sets are affine spaces in ℝ^{n} (see [21]). For reasons which will be explained in the final part of §2.3, our results are not immediately applicable to this situation. In a forthcoming paper [13], we will treat this aspect and related problems.

1.4 The basic example: Bad

Let I = [0, 1] and consider the set

Bad_{I} := {x ∈ [0, 1] : |x − p/q| > c(x)/q^{2} for all rationals p/q (q > 0)} .

This is the classical set Bad of badly approximable numbers restricted to the unit interval. Clearly, it can be expressed in the form Bad^{∗}(R, β, ρ) with ρ(r) := r^{−2} and

X = Ω := [0, 1] , J := {(p, q) ∈ ℕ × ℕ\{0} : p ≤ q} ,
α := (p, q) ∈ J , β_{α} := q , R_{α} := p/q .

The metric d is of course the standard Euclidean metric; d(x, y) := |x − y|. Thus in this basic example, the resonant sets R_{α} are simply rational points p/q and the function ρ clearly satisfies condition (B). With reference to our framework, let the measure m be one–dimensional Lebesgue measure on I. Thus, δ = 1 and m clearly satisfies condition (A).

We show that the conditions of Theorem 1 are satisfied for this basic example. The existence of the collection C(θB_{n}), where B_{n} is an arbitrary closed interval of length 2 k^{−2n} follows immediately from the following simple observation. For any two distinct rationals p/q and p′/q′ with k^{n} ≤ q, q′ < k^{n+1} we have that

|\frac{p}{q} − \frac{p′}{q′} | ≥ \frac{1}{qq′} > k^{−2n−2} .

Thus, any interval θB_{n} with θ := \frac{1}{2}k^{−2} contains at most one rational p/q with k^{n} ≤ q < k^{n+1}. Let C(θB_{n}) denote the collection of intervals 2θB_{n+1} obtained by subdividing θB_{n} into intervals of length 2k^{−2n−4} starting from the left hand side of θB_{n}. Clearly

#C(θB_{n}) ≥ [k^{2}/2] > k^{2}/4 = r.h.s. of (1) with κ_{1} := 1/4 .

Also, in view of the above observation, for k sufficiently large

l.h.s. of (2) ≤ 1 < k^{2}/8 = r.h.s. of (2) with κ_{2} := 1/8 .

The upshot of this is that Theorem 1 implies that

dim Bad_{I} = 1 .

In turn, since Bad is a subset of ℝ, this implies that dim Bad = 1 – the classical result of Jarník [10].

2 A more general framework

We now consider a more general framework in which the ‘badly approximable’ set consists of points avoiding ‘rectangular’ neighborhoods of resonant sets rather than simply ‘balls’.

Let (X, d) be the product space of t metric spaces (X_{i}, d_{i}) and let (Ω, d) be a compact subspace of X which contains the support of a non-atomic finite measure m. As before, let R = {R_{α} ∈ X : α ∈ J} be a family of subsets R_{α} of X indexed by an infinite, countable set J. Thus, each resonant set R_{α} can be split into its t components R_{α,i} ⊂ (X_{i}, d_{i}). As before, let β : J → ℝ^{+} : α → β_{α} be a positive function on J and assume that the number of α ∈ J with β_{α} bounded above is finite.

For each 1 ≤ i ≤ t, let ρ_{i} : ℝ^{+} → ℝ^{+} : r → ρ_{i}(r) be a real, positive function such that ρ_{i}(r) → 0 as r → ∞ and that ρ_{i} is decreasing for r large enough. Furthermore, assume that ρ_{1}(r) ≥ ρ_{2}(r) ≥ ⋯ ≥ ρ_{t}(r) for r large – the ordering is irrelevant. Given a resonant set R_{α}, let

F_{α}(ρ_{1}, …, ρ_{t}) := {x ∈ X : d_{i}(x_{i}, R_{α,i}) ≤ ρ_{i}(β_{α}) for all 1 ≤ i ≤ t} ,

denote the ‘rectangular’ (ρ_{1}, …, ρ_{t})-neighborhood of R_{α} and consider the set

Bad^{∗}(R, β, ρ_{1}, …, ρ_{t}) := {x ∈ Ω : ∃ c(x) > 0 such that x ∉ c(x) F_{α}(ρ_{1}, …, ρ_{t}) for all α ∈ J} .

Thus, x ∈ Bad^{∗}(R, β, ρ_{1}, …, ρ_{t}) if there exists a constant c(x) > 0 such that for all α ∈ J,

d_{i}(x_{i}, R_{α,i}) ≥ c(k) ρ_{i}(β_{α}) (1 ≤ i ≤ t) .

Clearly, Bad^{∗}(R, β, ρ_{1}, … ρ_{t}) is precisely the set Bad^{∗}(R, β, ρ) of §1.1 in the case t = 1. The overall aim of this section is to find a suitably general framework which gives a lower bound for the Hausdorff dimension of Bad^{∗}(R, β, ρ_{1}, …, ρ_{t}). We shall assume that sup_{α∈J} ρ_{i}(β_{α}) is finite for each i without loss of generality – otherwise Bad^{∗}(R, β, ρ_{1}, …, ρ_{t}) = ∅ and there is nothing to prove.

2.1 The conditions on the general framework

Given l_{1}, …, l_{t} ∈ ℝ^{+} and c ∈ Ω let

F (c; l_{1}, …, l_{t}) := {x ∈ X : d_{i}(x_{i}, c_{i}) ≤ l_{i} for all 1 ≤ i ≤ t} ,

denote the closed ‘rectangle’ centred at c with ‘sidelengths’ determined by l_{1}, …, l_{t}. Also, for any k > 1 and n ∈ ℕ, let F_{n} denote a generic rectangle F (c; ρ_{1}(k^{n}), …, ρ_{t}(k^{n}))∩Ω in Ω centred at a point c in Ω. As before, B(c, r) is a closed ball with centre c and radius r. The following conditions on the measure m and the functions ρ_{i} will play a central role in our general framework. The first two are reminiscent of conditions (A) and (B) of §1.2.

(A*) There exists a strictly positive constant δ such that for any c ∈ Ω

lim inf_{r→0} \frac{log m(B(c, r))}{log r} = δ .

It is easily verified that if the measure m supported on Ω is of type (A*) then dim Ω ≥ δ [6, Proposition 4.9] and so dim X ≥ δ. Clearly condition (A) of §1.2 implies (A*).

(B*) For k > 1 sufficiently large, any integer n ≥ 1 and any i ∈ {1, …, t},

λ_{i}^{l}(k) ≤ \frac{ρ_{i}(k^{n})}{ρ_{i}(k^{n+1})} ≤ λ_{i}^{u}(k),

where λ_{i}^{l} and λ_{i}^{u} are lower and upper constants such that λ_{i}^{l}(k) → ∞ as

k → ∞.

Clearly, this is just condition (B) of §1.2 imposed on each function ρ_{i}.

(C*) There exist constants 0 < a ≤ 1 ≤ b and l_{0} > 0 such that

a ≤ \frac{m(F (c; l_{1}, …, l_{t}))}{m(F (c′; l_{1}, …, l_{t}))} ≤ b,

for any c, c′ ∈ Ω and any l_{1}, …, l_{t} ≤ l_{0}.

This condition implies that rectangles of the same size centred at points of Ω have comparable m measure.

(D*) There exist strictly positive constants D and l_{0} such that

\frac{m(2 F (c; l_{1}, …, l_{t}))}{m(F (c; l_{1}, …, l_{t}))} ≤ D,

for any c ∈ Ω and any l_{1}, …, l_{t} ≤ l_{0}.

This condition simply says that the measure m is ‘doubling’ with respect to rectangles. In terms of achieving our aim of obtaining a lower bound for dim Bad^{∗}(R, β, ρ_{1}, … ρ_{t}), the above four conditions are rather natural. The following final condition is in some sense the only genuine technical condition and is not particularly restrictive.

(E*) For k > 1 sufficiently large and any integer n ≥ 1

\frac{m(F_{n})}{m(F_{n+1})} ≥ λ(k) ,

where λ is a constant such that λ(k) → ∞ as k → ∞.

2.2 The general result

Recall, that F_{n} := {x ∈ Ω : d_{i}(x_{i}, c_{i}) ≤ ρ_{i}(k^{n}) for all 1 ≤ i ≤ t} is a generic rectangle with centre c in Ω and ‘sidelengths’ determined by ρ_{i}(k^{n}) and for θ ∈ ℝ^{+}, θF_{n} is the rectangle F_{n} scaled by θ. Also, for n ≥ 1 let J(n) := {α ∈ J : k^{n−1} ≤ β_{α} < k^{n}}.

Theorem 2 Let (X, d) be the product space of the metric spaces (X_{1}, d_{1}), …, (X_{t}, d_{t}) and let (Ω, d, m) be a compact measure subspace of X. Let the measure m and the functions ρ_{i} satisfy conditions (A*) to (E*). For k ≥ k_{0} > 1, suppose there exists some θ ∈ ℝ^{+} so that for n ≥ 1 and any rectangle F_{n} there exists a disjoint collection C(θF_{n}) of rectangles 2θF_{n+1} contained within θF_{n} satisfying

#C(θF_{n}) ≥ κ_{1} \frac{m(θF_{n})}{m(θF_{n+1})}  (3)

and

# {2θF_{n+1} ⊂ C(θF_{n}) : min_{α∈J(n+1)} d_{i}(c_{i}, R_{α,i}) ≤ 2θρ_{i}(k^{n+1}) for any 1 ≤ i ≤ t}

≤ κ_{2} \frac{m(θF_{n})}{m(θF_{n+1})} . (4)

where 0 < κ_{2} < κ_{1} are absolute constants independent of k and n. Furthermore, suppose dim(∪_{α∈J}R_{α}) < δ. Then

dim Bad^{∗}(R, β, ρ_{1}, … ρ_{t}) ≥ δ .

Remarks: For k sufficiently large, it is always possible to find the collection C(θF_{n}) satisfying condition (3). Clearly, the lower bound for dim Bad^{∗}(R, β, ρ) in Theorem 1 is an immediate consequence of Theorem 2. To see this, simply note that if t = 1 then the rectangles F_{n} are balls B_{n} and if conditions (A) and (B) are satisfied then trivially so are the conditions (A*) to (E*). In fact, if condition (A*) is replaced by the stronger condition (A) in the above theorem, then we are able to conclude that dim Bad(R, β, ρ_{1}, … ρ_{t}) = δ – see below.

We now consider an extremely useful specialization of the above general framework in which the space Ω is a product space equipped with a product measure.

Theorem 3 For 1 ≤ i ≤ t, let (X_{i}, d_{i}) be a metric space and (Ω_{i}, d_{i}, m_{i}) be a compact measure subspace of X_{i} where the measure m_{i} satisfies condition (A) with exponent δ_{i}. Let (X, d) be the product space of the spaces (X_{i}, d_{i}) and let (Ω, d, m) be the product measure space of the measure spaces (Ω_{i}, d_{i}, m_{i}). Let the functions ρ_{i} satisfy condition (B*). For k ≥ k_{0} > 1, suppose there exists some θ ∈ ℝ^{+} so that for n ≥ 1 and any rectangle F_{n} there exists a disjoint collection C(θF_{n}) of rectangles 2θF_{n+1} contained within θF_{n} satisfying

#C(θF_{n}) ≥ κ_{1} ∏_{i=1}^{t} (\frac{ρ_{i}(k^{n})}{ρ_{i}(k^{n+1})})^{δ_{i}}  (5)

and

# {2θF_{n+1} ⊂ C(θF_{n}) : min_{α∈J(n+1)} d_{i}(c_{i}, R_{α,i}) ≤ 2θρ_{i}(k^{n+1}) for any 1 ≤ i ≤ t}

≤ κ_{2} ∏_{i=1}^{t} (\frac{ρ_{i}(k^{n})}{ρ_{i}(k^{n+1})})^{δ_{i}} , (6)

where 0 < κ_{2} < κ_{1} are absolute constants independent of k and n. Furthermore, suppose dim(∪_{α∈J}R_{α}) < ∑_{i=1}^{t} δ_{i}. Then

dim Bad^{∗}(R, β, ρ_{1}, … ρ_{t}) = ∑_{i=1}^{t} δ_{i} .

The deduction of Theorem 3 from Theorem 2 is relatively straightforward and hinges on the following simple observation. Since m is the product measure of the measures m_{i} and the latter satisfy condition (A) with exponents δ_{i} (1 ≤ i ≤ t), we have for any c ∈ Ω and any l_{1}, …, t_{t} ≤ l_{0} that

a^{t} ≤ \frac{m(F (c; l_{1}, …, l_{t}))}{∏_{i=1}^{t} l_{i}^{δ_{i}}} ≤ b^{t}.  (7)

It follows that conditions (C*) and (D*) are trivially satisfied as is condition (A) with δ := ∑_{i=1}^{t} δ_{i}. Recall, that (A) implies (A*). Also, (7) together with (B*) implies that condition (E*) is satisfied. Thus, Theorem 2 implies the desired lower bound estimate for dim Bad^{∗}(R, β, ρ_{1}, … ρ_{t}). The complementary upper bound estimate is a simple consequence of the fact that m satisfies (A). If m satisfies (A), then dim Ω = δ [6, Proposition 4.9] and since Bad^{∗}(R, β, ρ_{1}, …, ρ_{t}) ⊆ Ω the upper bound follows.

2.3 The general basic example: Bad(i, j)

For i, j ≥ 0 with i + j = 1, denote by Bad(i, j) the set of (i, j)-badly approximable pairs (x_{1}, x_{2}) ∈ ℝ^{2}; that is (x_{1}, x_{2}) ∈ Bad(i, j) if there exists a positive constant c(x_{1}, x_{2}) such that for all q ∈ ℕ

max{ ||qx_{1}||^{1/i} , ||qx_{2}||^{1/j} } > c(x_{1}, x_{2}) q^{−1} ,

where || · || denotes the distance of a real number to the nearest integer. In the case i = j = 1/2, the set under consideration is simply the standard set of badly approximable pairs. If i = 0 we identify the set Bad(0, 1) with ℝ×Bad where Bad is the set of badly approximable numbers. That is, Bad(0, 1) consists of pairs (x_{1}, x_{2}) with x_{1} ∈ ℝ and x_{2} ∈ Bad. The roles of x_{1} and x_{2} are reversed if j = 0. Recently [18], it has been shown that dim Bad(i, j) = 2. We now show that this result is in fact a simple consequence of Theorem 3.

Let Bad_{I^{2}}(i, j) := Bad(i, j) ∩ I^{2} where I^{2} := [0, 1] × [0, 1]. Without loss of generality assume that i ≤ j. Clearly, it can be expressed in the form Bad^{∗}(R, β, ρ_{1}, ρ_{2}) with ρ_{1}(r) := r^{−(1+i)}, ρ_{2}(r) := r^{−(1+j)} and

X = Ω := I^{2} , J := {((p_{1}, p_{2}), q) ∈ ℕ^{2} × ℕ\{0} : p_{1}, p_{2} ≤ q} ,
α := ((p_{1}, p_{2}), q) ∈ J , β_{α} := q , R_{α} := (p_{1}/q, p_{2}/q) .

Furthermore, d_{1} = d_{2} is the standard Euclidean metric on I and m_{1} = m_{2} is one–dimensional Lebesgue measure on I. By definition, the metric d on I^{2} is the product metric d_{1} × d_{1} and the measure m := m_{1} × m_{1} is simply two–dimensional Lebesgue measure on I^{2}.

We show that the conditions of Theorem 3 are satisfied for this basic example. Clearly the functions ρ_{1}, ρ_{2} satisfy condition (B*) and the measures m_{1}, m_{2} satisfy condition (A) with δ_{1} = δ_{2} = 1. We now need to establish the existence of the collection C(θF_{n}), where F_{n} is an arbitrary closed rectangle of size 2k^{−n(1+i)} × 2k^{−n(1+j)}. To start with, note that m(θF_{n}) = 4θ^{2}k^{−3n}. Now assume there are at least three rational points (p_{1}/q, p_{2}/q), (p′_{1}/q′, p′_{2}/q′) and (p′′_{1}/q′′, p′′_{2}/q′′) with

k^{n} ≤ q, q′, q′′ < k^{n+1}

lying within θF_{n}. Suppose for the moment that they do not lie on a line and form the triangle Δ sub-tended by them. Twice the area of the triangle Δ is equal to the absolute value of the determinant

det := |\begin{matrix}1 & p_{1}/q & p_{2}/q \\ 1 & p′_{1}/q′ & p′_{2}/q′ \\ 1 & p′′_{1}/q′′ & p′′_{2}/q′′ \end{matrix}| .

Then, in view of the denominator constraint, it follows that

2 × m(Δ) ≥ \frac{1}{qq′q′′} > k^{−3(n+1)} .

Now put

θ := 2^{−1}(2k^{3})^{−1/2}.

Then m(Δ) > m(θF_{n}) and this is impossible since Δ ⊂ θF_{n}. The upshot of this is that the triangle in question can not exist. Thus, if there are two or more rational points with k^{n} ≤ q < k^{n+1} lying within θF_{n} then they must lie on a line L.

Starting from a ‘corner’ of the rectangle θF_{n}, partition θF_{n} into rectangles 2θF_{n+1} of size 4k^{−(n+1)(1+i)} ×4k^{−(n+1)(1+j)} and denote by C(θF_{n}) the collection of rectangles 2θF_{n+1} obtained. Trivially

#C(θF_{n}) ≥ [\frac{2θk^{−n(1+i)}}{4θk^{−(n+1)(1+i)}} ] [\frac{2θk^{−n(1+j)}}{4θk^{−(n+1)(1+j)}} ] ≥ \frac{k^{3}}{16} .

In view of the above ‘triangle’ argument we have that

# {2θF_{n+1} ⊂ C(θF_{n}) : min_{α∈J(n+1)} d_{i}(c_{i}, R_{α,i}) ≤ 2θρ_{i}(k^{n+1}) for all 1 ≤ i ≤ t}

≤ # {2θF_{n+1} ⊂ C(θF_{n}) : 2θF_{n+1} ∩ L ≠ ∅} ,

where L is any line passing through θF_{n}. Recall, that we are assuming that i ≤ j. A simple geometric argument ensures that for k sufficiently large

# \{2θF_{n+1} ⊂ C(θF_{n}) : 2θF_{n+1} ∩ L ≠ ∅\} ≤ [\frac{2θk^{−n(1+j)}}{4θk^{−(n+1)(1+j)}} ] = [\frac{k^{1+j}}{2} ]
≤ k^{1+j} ≤ k^{3}/32 .

The upshot of this is that the collection C(θF_{n}) satisfies the required conditions and Theorem 3 implies that

dim Bad_{I^{2}}(i, j) = 2 .

In turn, since Bad(i, j) is a subset of ℝ^{2}, this implies that dim Bad(i, j) = 2.

In [18], the stronger result that dim Bad(i, j) ∩ Bad(1, 0) ∩ Bad(0, 1) = 2 is established; i.e. the set of pairs (x_{1}, x_{2}) with x_{1} and x_{2} both badly approximable numbers and an (i, j)-badly approximable pair has full dimension. In §5.1, we obtain a much more general result and remark on a beautiful conjecture of W.M. Schmidt. In full generality, Schmidt’s conjecture states that Bad(i, j) ∩ Bad(i′, j′) ≠ ∅. It is a simple exercise to show that if Schmidt’s conjecture is false for some pairs (i, j) and (i′, j′) then Littlewood’s conjecture in simultaneous Diophantine approximation is true.

We now turn our attention to the natural generalization of Bad(i, j) to higher dimensions. For any N-tuple of real numbers i_{1}, …, i_{N} ≥ 0 such that ∑ i_{r} = 1, denote by Bad(i_{1}, …, i_{N}) the set of points (x_{1}, …, x_{N}) ∈ ℝ^{N} for which there exists a positive constant c(x_{1}, …, x_{N}) such that for any q ∈ ℕ,

max{ ||qx_{1}||^{1/i_{1}} , …, ||qx_{N}||^{1/i_{N}} } > c(x_{1}, …, x_{N}) q^{−1}.

Clearly, the two-dimensional argument can easily be modified to show that

dim Bad(i_{1}, …, i_{N}) = N .

The key modification is the following lemma which naturally extends the main feature of the ‘triangle’ argument in dimension two to a ‘simplex’ one in dimension N.

Lemma 4 (Simplex Lemma) Let N ≥ 1 be an integer and k > 1 be a real number. Let E ⊆ ℝ^{N} be a convex set of N-dimensional Lebesgue measure

|E| ≤ (N! )^{−1}k^{−(N+1)} .

Suppose that E contains N + 1 rational points (p_{i}^{(1)}/q_{i}, …, p_{i}^{(N)}/q_{i}) with 1 ≤ q_{i} < k, where 0 ≤ i ≤ N. Then these rational points lie in some hyperplane.

Proof. Suppose to the contrary that this is not the case. In that case, the rational points (p_{i}^{(1)}/q_{i}, …, p_{i}^{(N)}/q_{i}) where 0 ≤ i ≤ N are distinct. Consider the N-dimensional simplex Δ subtended by them; i.e. an interval when N = 1, a triangle when N = 2, a tetrahedron when N = 3 and so on. Clearly, Δ is a subset of E since E is convex. The volume of the simplex |Δ| times N factorial is equal to the absolute value of the determinant

det := |\begin{matrix}1 & p_{0}^{(1)}/q_{0} & ⋯ & p_{0}^{(N)}/q_{0} \\ 1 & p_{1}^{(1)}/q_{1} & ⋯ & p_{1}^{(N)}/q_{1} \\ ⋮ & ⋮ & & ⋮ \\ 1 & p_{N}^{(1)}/q_{N} & ⋯ & p_{N}^{(N)}/q_{N} \end{matrix} | .

As this determinant is not zero, it follows from the assumption made on the q_{i} that

N! × |Δ| = | det | ≥ \frac{1}{q_{0}q_{1} ⋯ q_{N}} > k^{−(N+1)} .

Consequently, |Δ| > (N! )^{−1}k^{−(N+1)} ≥ |E|. This contradicts the fact that Δ ⊆ E. □

Remarks:

(i) The Simplex Lemma should be viewed as the higher dimensional generalization of the following simple fact already exploited in the §1.4: on the real line ℝ an interval I_{k} of length 1/k^{2} can contain at most one rational p/q with 1 ≤ q < k. This follows from the trivial observation that if 1 ≤ q, q′ < k then |p/q − p′/q′| ≥ 1/qq′ > 1/k^{2}.

(ii) Our general setup will be applied to settings other than subsets of ℝ^{N} (see §5). In most of these, an analogue of the Simplex Lemma will be required. In these settings we will either give a complete proof or sketch the argument required in two dimensions; i.e. the analogue of the ‘triangle’ argument. Based on the proof of the Simplex Lemma in ℝ^{N}, it should then be obvious how to extend the N = 2 argument to higher dimensions. In short, within this paper the main ideas are always exposed on establishing a given N-dimensional statement in the N = 2 case. The proof in higher dimensions requires no new ideas. Thus in all the various applications of our general framework, for the sake of both clarity and notation we shall stick to N = 2 in proofs.

(iii) The ‘triangle’ argument (or variants thereof) described above is critical in most of the applications considered in this paper (see §5). To some extent this is the reason why our main results cannot be directly applied to the problem of badly approximable systems of linear forms. In this case the resonant sets R_{α} are affine spaces and although the ‘triangle’ or more generally the ‘simplex’ approach remains the main ingredient it requires deeper considerations in the geometry of numbers to successfully execute it. We will return to this and other aspects of the linear forms theory in a forthcoming paper [13].

3 Preliminaries

In this short section we define Hausdorff measure and dimension in order to establish some notation and then describe a method for obtaining lower bounds for the dimension.

Suppose Ω is a non–empty subset of (X, d). For ρ > 0, a countable collection {B_{i}} of balls in X with radii r_{i} ≤ ρ for each i such that Ω ⊂ ⋃_{i} B_{i} is called a ρ-cover for Ω. Clearly such a cover always exists for totally bounded metric spaces. Let s be a non-negative number and define

H_{ρ}^{s}(Ω) = inf {∑_{i} r_{i}^{s} : {B_{i}} is a ρ-cover of Ω} ,

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

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

and the Hausdorff dimension dim Ω of a set Ω by

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

In particular when s is an integer H^{s} is comparable to s-dimensional Lebesgue measure. For further details see [6,16]. A general and classical method for obtaining a lower bound for the Hausdorff dimension of an arbitrary set Ω is the following mass distribution principle (see e.g. [6, page 55]).

Lemma 5 (Mass Distribution Principle) Let μ be a probability measure supported on a subset Ω of (X, d). Suppose there are positive constants c and r_{0} such that

μ(B) ≤ c r^{s} ,

for any ball B with radius r ≤ r_{0} . Then H^{s}(Ω) ≥ 1/c. In particular, we have that dim Ω ≥ s.

The following rather simple covering result will be crucial to our proof of Theorem 2.

Lemma 6 (Covering Lemma) Let (X, d) be the product space of the metric spaces (X_{1}, d_{1}), …, (X_{t}, d_{t}) and F be a finite collection of ‘rectangles’ F := F (c; l_{1}, …, l_{t}) with c ∈ X and l_{1}, …, l_{t} fixed. Then there exists a disjoint sub-collection {F_{m}} such that

⋃_{F ∈F} F ⊂ ⋃_{m} 3F_{m} .

Proof. Let S denote the set of centres c of the rectangles in F. Choose c(1) ∈ S and for k ≥ 1,

c(k + 1) ∈ S \ ⋃_{m=1}^{k} 2 F (c(m); l_{1}, …, l_{t})

as long as S \ ⋃_{m=1}^{k} 2 F (c(m); l_{1}, …, l_{t}) ≠ ∅. Since #S is finite, there exists k_{1} ≤ #S such that

S ⊂ ⋃_{m=1}^{k_{1}} 2 F (c(m); l_{1}, …, l_{t}) .

By construction, any rectangle F (c; l_{1}, …, l_{t}) in the original collection F is contained in some rectangle 3 F (c(m); l_{1}, …, l_{t}) and since d_{i}(c_{i}(m), c_{i}(n)) > 2l_{i} for each 1 ≤ i ≤ t the chosen rectangles F (c(m); l_{1}, …, l_{t}) are clearly disjoint.

□

We end this section by making use of the covering lemma to establish the following assertion made in §2.2. The result is extremely useful when it comes to applying our theorems – see §5. With reference to Theorem 2, it guarantees the existence of a disjoint collection C(θF_{n}) of rectangles with the necessary cardinality.

Lemma 7 Let (X, d) be the product space of the metric spaces (X_{1}, d_{1}), …, (X_{t}, d_{t}) and let (Ω, d, m) be a compact measure subspace of X. Let the measure m and the functions ρ_{i} satisfy conditions (B*) to (D*). Let k be sufficiently large. Then for any θ ∈ ℝ^{+} and for any rectangle F_{n} (n ≥ 1) there exists a disjoint collection C(θF_{n}) of rectangles 2θF_{n+1} contained within θF_{n} satisfying (3) of Theorem 2.

Proof. Begin by choosing k large enough so that for any i ∈ {1, …, t},

\frac{ρ_{i}(k^{n})}{ρ_{i}(k^{n+1})} ≥ 4.  (8)

That this is possible follows from the fact that λ_{i}^{l}(k) → ∞ as k → ∞ (condition (B*)). Take an arbitrary rectangle F_{n} and let l_{i}(n) := θρ_{i}(k^{n}). Thus θF_{n} := F (c; l_{1}(n), …, l_{t}(n)). Consider the rectangle T_{n} ⊂ θF_{n} where

T_{n} := F (c; l_{1}(n) − 2l_{1}(n + 1), …, l_{t}(n) − 2l_{t}(n + 1)) .

Note that in view of (8) we have that T_{n} ⊃ \frac{1}{2}θF_{n}. Now, cover T_{n} by rectangles 2θF_{n+1} with centres in Ω∩T_{n}. By construction, these rectangles are contained in θF_{n} and in view of the covering lemma there exists a disjoint sub-collection C(θF_{n}) such that

T_{n} ⊂ ⋃_{2θF_{n+1}⊂C(θF_{n})} 6θF_{n+1} .

Using that fact that rectangles of the same size centred at points of Ω have comparable m measure (condition (C*)), it follows that

a m(\frac{1}{2}θF_{n}) ≤ m(T_{n}) ≤ #C(θF_{n}) b m(6θF_{n+1}) .

Using that fact that the measure m is doubling on rectangles (condition (D*)), so that m(\frac{1}{2}θF_{n}) ≥ D^{−1}m(θF_{n}) and m(6θF_{n+1}) ≤ m(8θF_{n+1}) ≤ D^{3}m(θF_{n+1}), it follows that

#C(θF_{n}) ≥ \frac{a}{bD^{4}} \frac{m(θF_{n})}{m(θF_{n+1})} .

□

Remark. Clearly, with reference to Theorem 1, the above lemma guarantees the existence of the collection C(θB_{n}) satisfying (1).

4 Proof of Theorem 2

The overall strategy is as follows. For any k sufficiently large we construct a Cantor-type set K_{c(k)} such that K_{c(k)} with at most a finite number of points removed is a subset of Bad^{∗}(R, β, ρ_{1}, …, ρ_{t}). Next, we construct a measure μ supported on K_{c(k)} with the property that for any ball A with radius r(A) sufficiently small

μ(A) ≪ r(A)^{δ−ϵ(k)} ;

where ϵ(k) → 0 as k → ∞. Hence, by construction and the mass distribution principle we have that

dim Bad^{∗}(R, β, ρ_{1}, …, ρ_{t}) ≥ dim K_{c(k)} ≥ δ − ϵ(k) .

Now suppose that dim Bad^{∗}(R, β, ρ_{1}, …, ρ_{t}) < δ. Then, dim Bad^{∗}(R, β, ρ_{1}, …, ρ_{t}) = δ − η for some η > 0. However, by choosing k large enough so that ϵ(k) < η we obtain a contradiction and thereby the lower bound result follows.

4.1 The Cantor-type set K_{c(k)}

Choose k_{0} sufficiently large so that for k ≥ k_{0}, ρ_{i}(k) (1 ≤ i ≤ t) is decreasing and the hypotheses of the theorem are valid. Now fix some k ≥ k_{0} and suppose that

{α ∈ J : β_{α} < k} = ∅ .  (9)

Define F_{1} to be any rectangle θF_{1} of radius θρ(k) and centre c in Ω. The idea is to establish, by induction on n, the existence of a collection F_{n} of disjoint rectangles θF_{n} such that F_{n} is nested in F_{n−1}; that is, each rectangle θF_{n} in F_{n} is contained in some rectangle θF_{n−1} of F_{n−1}. Also, any θF_{n} in F_{n} will have the property that for all points x ∈ θF_{n}, for all i ∈ {1, …, t} and for all α ∈ J with β_{α} < k^{n},

d_{i}(x, R_{α,i}) ≥ c(k) ρ_{i}(β_{α}),  (10)

where the constant

c(k) := min_{1≤i≤t}(θ/λ_{i}^{u}(k))

is dependent on k but is independent of n. Then, since the rectangles θF_{n} of F_{n} are closed, nested and the space Ω is compact, any limit point in θF_{n} will satisfy (10) for all α in J with β_{α} ≥ k. In particular, we put

K_{c(k)} := ⋂_{n=1}^{∞} F_{n} .

By construction, we have that K_{c(k)} is a subset of Bad^{∗}(R, β, ρ_{1}, …, ρ_{t}) under the assumption (9).

The induction. For n = 1, (10) is trivially satisfied for F_{1} = θF_{1} since we are assuming (9). Given F_{n} satisfying (10) we wish to construct a nested collection F_{n+1} for which (10) is satisfied for n+1. Consider any rectangle θF_{n} ⊂ F_{n}. We construct a ‘local’ collection F_{n+1}(θF_{n}) of disjoint rectangles θF_{n+1} contained in θF_{n} so that for any point x ∈ θF_{n+1} the condition given by (10) is satisfied for n + 1. Given that any rectangle θF_{n+1} of F_{n+1}(θF_{n}) is to be nested in θF_{n}, it is enough to show that for any point x ∈ θF_{n+1} the inequalities

d_{i}(x_{i}, R_{α,i}) ≥ c(k) ρ_{i}(β_{α}) (1 ≤ i ≤ t)

are satisfied for α ∈ J with k^{n} ≤ β_{α} < k^{n+1}; i.e. with α ∈ J(n + 1).

For k sufficiently large, by the hypotheses of the theorem, there exists a disjoint sub-collection G(θF_{n}) of C(θF_{n}) of rectangles 2θF_{n+1} ⊂ θF_{n} with

#G(θF_{n}) = [κ \frac{m(θF_{n})}{m(θF_{n+1})}]  κ := min\{1, \frac{1}{2}(κ_{1} − κ_{2})\} , (11)

and such that for any rectangle 2θF_{n+1} ⊂ G(θF_{n}) with centre c

min_{α∈J(n+1)} d_{i}(c_{i}, R_{α,i}) ≥ 2 θ ρ_{i}(k^{n+1}) .

Clearly, by choosing k large enough we can ensure that #G(θF_{n}) > 1 – this makes use of conditions (D*) and (E*). Now let

F_{n+1}(θF_{n}) := {θF_{n+1} : 2θF_{n+1} ⊂ G(θF_{n})} .

Thus the rectangles of F_{n+1}(θF_{n}) are precisely those of G(θF_{n}) but scaled by a factor 1/2. Then, by construction for any x ∈ θF_{n+1} ⊂ F_{n+1}(θF_{n}) and 1 ≤ i ≤ t

d_{i}(x_{i}, R_{α,i}) ≥ θρ_{i}(k^{n+1}) = θρ_{i}(k^{n}) \frac{ρ_{i}(k^{n+1})}{ρ_{i}(k^{n})} ≥ \frac{θ}{λ_{i}^{u}(k)} ρ_{i}(β_{α})
≥ c(k) ρ_{i}(β_{α}).

Here we have made use of condition (B*) and the fact that ρ_{i}(k) is decreasing for k ≥ k_{0} and that α ∈ J(n + 1). Finally let

F_{n+1} := ⋃_{θF_{n}∈F_{n}} F_{n+1}(θF_{n}) .

This completes the proof of the induction step and so the construction of the Cantor-type set

K_{c(k)} := ⋂_{n=1}^{∞} F_{n} ,

where c(k) := min_{1≤i≤t}(θ/λ_{i}^{u}(k)) and k is sufficiently large.

Note, that in view of (11) we have that for n ≥ 2

#F_{n} = #F_{n−1} × #F_{n}(θF_{n−1}) = ∏_{m=2}^{n} #F_{m}(θF_{m−1})

≥ ∏_{m=2}^{n} \frac{κ}{2} \frac{m(θF_{m−1})}{m(θF_{m})} = (\frac{κ}{2})^{n−1} \frac{m(θF_{1})}{m(θF_{n})}. (12)

4.2 The measure μ on K_{c(k)}

We now describe a probability measure μ supported on the Cantor–type set K_{c(k)} constructed in the previous subsection. For any rectangle θF_{n} in F_{n} we attach a weight μ(θF_{n}) which is defined recursively as follows: for n = 1,

μ(θF_{1}) := \frac{1}{#F_{1}} = 1

and for n ≥ 2,

μ(θF_{n}) := \frac{1}{#F_{n}(θF_{n−1})} μ(θF_{n−1}) (F_{n} ⊂ F_{n−1}) .

This procedure thus defines inductively a mass on any rectangle used in the construction of K_{c(k)}. In fact a lot more is true – μ can be further extended to all Borel subsets A of Ω to determine μ(A) so that μ constructed as above actually defines a measure supported on K_{c(k)}; see [6, Proposition 1.7]. We state this formally as a

Fact. The probability measure μ constructed above is supported on K_{c(k)} and for any Borel subset A of Ω

μ(A) = inf ∑_{F ∈F} μ(F) .

The infimum is over all coverings F of A by rectangles F ∈ {F_{n} : n ≥ 1}.

Notice that, in view of (12), we simply have that

μ(θF_{n}) = \frac{1}{#F_{n}} (n ≥ 1) .

4.3 A lower bound for dim K_{c(k)}

Let A be an arbitrary ball with centre a not necessarily in Ω and of radius r(A) < θρ_{∗}(k^{n_{0}}) where ρ_{∗}(r) := max_{1≤i≤t} ρ_{i}(r) and n_{0} is to be determined later. We now determine an upper bound for μ(A) in terms of its radius. Choose n ≥ n_{0} so that

θρ_{∗}(k^{n+1}) < r(A) ≤ θρ_{∗}(k^{n}) .

Without loss of generality, assume that A ∩ K_{c(k)} ≠ ∅ since otherwise there is nothing to prove. Clearly

μ(A) ≤ N_{n+1}(A) × μ(θF_{n+1})

where

N_{n+1}(A) := #{θF_{n+1} ⊂ F_{n+1} : θF_{n+1} ∩ A ≠ ∅} .

If θF_{n+1} ∩ A ≠ ∅, then θF_{n+1} ⊂ 3 A since r(A) ≥ θρ_{i}(k^{n+1}) for 1 ≤ i ≤ t. The balls in F_{n+1} are disjoint and have comparable m measure (condition (C*)), thus

N_{n+1}(A) ≤ \frac{m(3A)}{a m(θF_{n+1})} .

It follows by (12), that

μ(A) ≤ \frac{m(3A)}{a m(θF_{n+1})} × \frac{1}{#F_{n+1}} ≤ \frac{m(3A)}{a m(θF_{1})} (\frac{2}{κ})^{n} .

Using the fact that ρ_{∗}(k^{n}) ≤ λ_{∗}^{l}(k)^{−(n−1)}ρ_{∗}(k), it is easily verified that

\frac{1}{a m(θF_{1})} (\frac{2}{κ})^{n} < (\frac{1}{θρ_{∗}(k^{n})})^{ϵ(k)}

for

n ≥ n_{1} := [4 + \begin{matrix}log & \frac{(θ ρ_{∗}(k))^{ϵ(k)}}{a m(θF_{1})} \\ log & \frac{2}{κ} \end{matrix} ] and ϵ(k) := \frac{4 log \frac{2}{κ}}{log λ_{∗}^{l}(k)} .

Hence,

μ(A) ≤ m(3A) × (θρ_{∗}(k^{n}))^{−ϵ(k)} .

Since A ∩ K_{c(k)} ≠ ∅, there exists some point x ∈ A ∩ Ω. Moreover, 3A ⊂ B(x, 4 r(A)) which together with condition (A*) implies that

m(3A) ≤ m(B(x, 4 r(A))) ≤ r(A)^{δ−ϵ(k)}

for r(A) ≤ r_{0} := r_{0}(ϵ(k)). Now ρ_{∗}(r) → 0 as r → ∞, so θρ_{∗}(k^{n}) < r_{0} for n ≥ n_{2}. Thus, for n ≥ n_{0} := max{n_{1}, n_{2}}

μ(A) ≤ r(A)^{δ−ϵ(k)} × (θρ_{∗}(k^{n}))^{−ϵ(k)} .

On using the fact that r(A) ≤ θρ_{∗}(k^{n}), we obtain that

μ(A) ≤ r(A)^{δ−2ϵ(k)} .

This together with the mass distribution principle implies that

dim K_{c(k)} ≥ δ − 2ϵ(k) .

Note that since ϵ(k) → 0 as k → ∞ we have that dim K_{c(k)} → δ as k → ∞.

4.4 Completion of proof

Recall, that dim(∪_{α∈J}R_{α}) < δ. Now suppose that

dim Bad^{∗}(R, β, ρ_{1}, …, ρ_{t}) < δ .

It follows that max{dim Bad^{∗}(R, β, ρ_{1}, …, ρ_{t}), dim(∪_{α∈J}R_{α})} = δ − η for some η > 0. Fix some k sufficiently large so that 2 ϵ(k) < η. Then,

dim K_{c(k)} ≥ δ − 2ϵ(k) > δ − η .

By construction, for any point x ∈ K_{c(k)} we have for all α ∈ J with β_{α} ≥ k that

d_{i}(x_{i}, R_{α,i}) ≥ c(k) ρ_{i}(β_{α}) (1 ≤ i ≤ t) .

Now let J_{k} := {α ∈ J : β_{α} < k}. If (9) is true for our fixed k then J_{k} = ∅ and clearly K_{c(k)} ⊆ Bad^{∗}(R, β, ρ_{1}, …, ρ_{t}). In turn, dim Bad^{∗}(R, β, ρ_{1}, …, ρ_{t}) ≥ dim K_{c(k)} > δ − η and we have a contradiction. So suppose, J_{k} ≠ ∅ and let R_{k} := {R_{α} : α ∈ J_{k}}. For any fixed k the number of elements in J_{k} is finite. So, if x ∉ R_{k} then there exists a constant c′(x) > 0 such that for all α ∈ J_{k},

d_{i}(x_{i}, R_{α,i}) ≥ c′(k) ρ_{i}(β_{α}) (1 ≤ i ≤ t) .

Thus, for x ∈ K_{c(k)}\ R_{k} and α ∈ J,

d_{i}(x_{i}, R_{α,i}) ≥ c^{∗}(k) ρ_{i}(β_{α}) (1 ≤ i ≤ t) ,

where c^{∗}(x) := min{c(k), c′(x)}. It follows that

Bad^{∗}(R, β, ρ_{1}, …, ρ_{t}) ⊇ K_{c(k)}\ R_{k} ,

and since dim R_{k} < dim K_{c(k)} we have that

dim Bad^{∗}(R, β, ρ_{1}, …, ρ_{t}) ≥ dim(K_{c(k)}\ R_{k})
= dim K_{c(k)} ≥ δ − 2ϵ(k) > δ − η .

This is a contradiction and completes the proof of Theorem 2. □

5 Applications

5.1 Intersecting sets with Bad(i_{1}, …, i_{N})

Let Bad(i_{1}, …, i_{N}) be the set of (i_{1}, …, i_{N})-badly approximable N-tuples in ℝ^{N} as defined in §2.3 and Bad(N) := Bad(i_{1}, …, i_{N}) with i_{1} = … = i_{N} = 1/N. Thus Bad(1) is simply the set Bad of badly approximable real numbers. Let Ω be a compact subset of ℝ^{N}. The problem is to determine conditions on Ω under which

Bad_{Ω}(i_{1}, …, i_{N}) := Ω ∩ Bad(i_{1}, …, i_{N})

is of full dimension; i.e. dim Bad_{Ω}(i_{1}, …, i_{N}) = dim Ω. Recall, that the ‘2dimensional’ argument of §2.3 can easily be extended to show that dim Bad(i_{1}, …, i_{N}) = N.

To begin with, we address the above problem for the set Bad_{Ω}(N) = Ω ∩ Bad(N) in the case that Ω supports an ‘absolutely α-decaying’ measure that satisfies condition (A).

The notion of an ‘absolutely decaying’ measure was introduced in [11]. The following restrictive definition, exploited in [19], serves our purpose. Let Ω be a compact subset of ℝ^{N} which supports a non-atomic, finite measure m. Let L denote a generic hyperplane of ℝ^{N} and let L^{(ϵ)} denote its ϵ-neighborhood. We say that m is absolutely α-decaying if there exist strictly positive constants C, α, r_{0} such that for any hyperplane L, any ϵ > 0, any x ∈ Ω and any r < r_{0},

m (B(x, r) ∩ L^{(ϵ)}) ≤ C (\frac{ϵ}{r} )^{α} m(B(x, r)) .

In the case N = 1, the hyperplane L is simply a point a ∈ ℝ and L^{(ϵ)} is the ball B(a, ϵ) centred at a of radius ϵ. Also note that in this case, if the measure m satisfies condition (A) with exponent δ then m is automatically absolutely δ-decaying.

Theorem 8 Let Ω be a compact subset of ℝ^{N} which supports a measure m satisfying condition (A) and which in addition is absolutely α-decaying for some α > 0. Then

dim Bad_{Ω}(N) = dim Ω .

Proof. With reference to §1, the set Bad_{Ω}(N) can be expressed in the form Bad^{∗}(R, β, ρ) with ρ(r) := r^{−(1+ \frac{1}{N})} and

X = (ℝ^{N}, d) , J := {((p_{1}, …, p_{N}), q) ∈ ℕ^{N} × ℕ\{0}} ,
α := ((p_{1}, …, p_{N}), q) ∈ J , β_{α} := q , R_{α} := (p_{1}/q, …, p_{N}/q) .

Here d is standard sup metric on ℝ^{N}; d(x, y) := max{d(x_{1}, y_{1}), …, d(x_{N}, y_{N})}. Thus balls B(c, r) in ℝ^{N} are genuinely cubes of sidelength 2r.

We show that the conditions of Theorem 1 are satisfied. Clearly the function ρ satisfies condition (B) and we are given that the measure m supported on Ω satisfies condition (A). Also, since the resonant sets are points the condition that dim(∪_{α∈J}R_{α}) < δ is satisfied. We need to establish the existence of the disjoint collection C(θB_{n}) of balls (cubes) 2θB_{n+1} where B_{n} is an arbitrary

ball of radius k^{−n(1+ \frac{1}{N} )} with centre in Ω. In view of Lemma 7, there exists a disjoint collection C(θB_{n}) such that

#C(θB_{n}) ≥ κ_{1} k^{(1+ \frac{1}{N})δ} ;  (13)

i.e. (1) of Theorem 1 holds. We now verify that (2) is satisfied for any such collection.

We consider two cases.

Case 1: N = 1. The trivial argument of §1.4 shows that any interval θB_{n} with θ := \frac{1}{2}k^{−2} contains at most one rational p/q with k^{n} ≤ q < k^{n+1}; i.e. α ∈ J(n + 1). Thus, for k sufficiently large

l.h.s. of (2) ≤ 1 < \frac{1}{2} × r.h.s. of (13) .

Hence (2) is trivially satisfied and Theorem 1 implies the desired result.

Case 2: N ≥ 2. We shall prove the theorem in the case that N = 2. There are no difficulties and no new ideas are required in extending the proof to higher dimensions, especially in view of the Simplex Lemma (see §2.3).

Suppose that there are three or more rational points (p_{1}/q, p_{2}/q) with k^{n} ≤ q < k^{n+1} lying within the ball/square θB_{n}. Now put θ := 2^{−1}(2k^{3})^{−1/2}. Then the ‘triangle’ argument of §2.3 (where m is Lebesgue measure) implies that the rational points must lie on a line L passing through θB_{n}. It follows that

l.h.s. of (2) ≤ # {2θB_{n+1} ⊂ C(θB_{n}) : 2θB_{n+1} ∩ L ≠ ∅}

≤ # \{2θB_{n+1} ⊂ C(θB_{n}) : 2θB_{n+1} ⊂ L^{(ϵ)}\} for ϵ := 8θk^{−(n+1) \frac{3}{2}}

≤ \frac{m(θB_{n} ∩ L^{(ϵ)})}{m(2θB_{n+1})} the balls 2θB_{n+1} are disjoint

≤ a^{−1}b C 8^{α} 2^{−δ} k^{\frac{2}{3} (δ−α)} m is absolutely α-decaying

< \frac{1}{2} × r.h.s. of (13) for k sufficiently large.

Hence (2) is satisfied and Theorem 1 implies the desired result. □

The following statement which combines Theorems 2.2 and 8.1 of [11], shows that a large class of fractal measures are absolutely α-decaying and satisfy condition (A).

Theorem 9 Let {S_{1}, …, S_{k}} be an irreducible family of contracting self similarity maps of ℝ^{N} satisfying the open set condition and let m be the restriction of H^{δ} to its attractor K where δ := dim K. Then m is absolutely α-decaying and satisfies condition (A).

The simplest examples of such sets include regular Cantor sets, the Sierpiński gasket and the von Koch curve. All the terminology except for ‘irreducible’ is pretty much standard – see for example [6, Chp.9]. The notion of irreducible introduced in [11, §2] avoids the natural obstruction that there is a finite collection of proper affine subspaces of ℝ^{N} which is invariant under {S_{1}, …, S_{k}}. More recently, the class of examples regarding absolutely α-decaying measures has been extended by Urbański [23,24].

In view of Theorem 9, the following statement is a simple consequence of Theorem 8. It has also been independently established by Kleinbock & Weiss [11, Theorem 10.3] and [12]. In fact, Theorem 8 is also derived in [12] by an alternative approach.

Corollary 10 Let {S_{1}, …, S_{k}} be an irreducible family of contracting self similarity maps of ℝ^{N} satisfying the open set condition and let m be the restriction of H^{δ} to its attractor K where δ := dim K. Then

dim(K ∩ Bad(N)) = dim K .

We now consider the more general problem of determining conditions on Ω under which dim Bad_{Ω}(i_{1}, …, i_{N}) = dim Ω. Under the hypotheses of Theorem 2, by modifying the definition of ‘absolutely decaying’ to accommodate ‘rectangles’ it is clearly possible to obtain an analogue of the ‘abstract’ theorem (Theorem 8) for Bad_{Ω}(i_{1}, …, i_{N}). We have decided against establishing such a statement in this paper. The reason for this is simple. We are currently unable to prove the existence of a natural class of sets satisfying the more general ‘rectangular’ hypotheses. Nevertheless, in the special case that Ω is a product space we are able to prove the following statement.

Theorem 11 For 1 ≤ j ≤ N, let Ω_{j} be a compact subset of ℝ which supports a measure m_{j} satisfying condition (A) with exponent δ_{j}. Let Ω denote the product set Ω_{1} × … × Ω_{N}. Then, for any N-tuple (i_{1}, …, i_{N}) with i_{j} ≥ 0 and ∑_{j=1}^{N} i_{j} = 1,

dim Bad_{Ω}(i_{1}, ..., i_{N}) = dim Ω .

A simple application of the above theorem leads to following result.

Corollary 12 Let K_{1} and K_{2} be regular Cantor subsets of ℝ. Then

dim ((K_{1} × K_{2}) ∩ Bad(i, j)) = dim(K_{1} × K_{2}) = dim K_{1} + dim K_{2} .

Proof of Theorem 11. Without loss of generality assume that N ≥ 2. The case that N = 1 is covered by Theorem 8. For the sake of clarity, as with the proof of Theorem 8, we shall restrict our attention to the case N = 2.

Recall that since Ω_{j} ⊂ ℝ and m_{j} satisfies (A), then m_{i} is automatically absolutely δ_{j}-decaying. A relatively straightforward argument shows that m := m_{1} × m_{2} is absolutely α-decaying on Ω with α := min{δ_{1}, δ_{2}}. In fact this trivially follows from the following general fact - see [11, §9].

Fact: For 2 ≤ j ≤ N, if each m_{j} is absolutely α_{j}-decaying on Ω_{j}, then m := m_{1} × … × m_{N} is absolutely α-decaying on Ω = Ω_{1} × … × Ω_{N} with α = min{α_{1}, …, α_{N}}.

Now let us write Bad(i, j) for Bad(i_{1}, i_{2}) and without loss of generality assume that i < j. The case i = j is already covered by Theorem 4 since m is absolutely α-decaying on Ω and clearly satisfies condition (A). The set Bad_{Ω}(i, j) can be expressed in the form Bad^{∗}(R, β, ρ_{1}, ρ_{2}) with ρ_{1}(r) = r^{−(1+i)}, ρ_{2}(r) = r^{−(1+j)} and

X = ℝ^{2} , Ω := Ω_{1} × Ω_{2} , J := {((p_{1}, p_{2}), q) ∈ ℕ^{2} × ℕ \ {0}} ,
α := ((p_{1}, p_{2}), q) ∈ J , β_{α} := q , R_{α} := (p_{1}/q, p_{2}/q) .

With reference to Theorem 3, the functions ρ_{1}, ρ_{2} satisfy condition (B*) and the measures m_{1}, m_{2} satisfy condition (A). Also note that dim(∪_{α∈J}R_{α}) = 0 since the union in question is countable. We need to establish the existence of the collection C(θF_{n}), where F_{n} is an arbitrary closed rectangle of size 2k^{−n(1+i)} × 2k^{−n(1+j)} with centre c in Ω. In view of Lemma 7, there exists a disjoint collection C(θF_{n}) of rectangles 2θF_{n+1} ⊂ θF_{n} such that

#C(θF_{n}) ≥ κ_{1} k^{(1+i)δ_{1}}k^{(1+j)δ_{2}};  (14)

i.e. (5) of Theorem 3 is satisfied. We now verify that (6) is satisfied for any such collection. With θ = 2^{−1}(2k^{3})^{−1/2}, the ‘triangle’ argument or equivalently the Simplex Lemma of §2.3 implies that

l.h.s. of (6) ≤ #{2θF_{n+1} ⊂ C(θF_{n}) : 2θF_{n+1} ∩ L ≠ ∅} ,  (15)

where L is a line passing through θF_{n}. Consider the thickening T (L) of L obtained by placing rectangles 4θF_{n+1} centred at points of L; that is, by ‘sliding’ a rectangle 4θF_{n+1}, centred at a point of L, along L. Then, since the rectangles 2θF_{n+1} ⊂ C(θF_{n}) are disjoint,

#{2θF_{n+1} ⊂ C(θF_{n}) : 2θF_{n+1} ∩ L ≠ ∅}

≤ #{2θF_{n+1} ⊂ C(θF_{n}) : 2θF_{n+1} ⊂ T (L)}

≤ \frac{m(T (L) ∩ θF_{n})}{m(2θF_{n+1})}.  (16)

Without loss of generality we can assume that L passes through the centre of θF_{n}. To see this, suppose that m(T (L) ∩ θF_{n}) ≠ 0 since otherwise there is nothing to prove. Then, there exists a point x ∈ T (L) ∩ θF_{n} ∩ Ω such that

T (L) ∩ θF_{n} ⊂ 2θF′_{n} ∩ T′(L′) .

Here F′_{n} is the rectangle of size k^{−n(1+i)} × k^{−n(1+j)} centred at x, L′ is the line parallel to L passing through x and T′(L′) is the thickening obtained by ‘sliding’ a rectangle 8θF_{n+1} centred at x, along L′. Then the following argument works just as well on 2θF′_{n} ∩ T′(L′).

Let Δ denote the slope of the line L and assume that Δ ≥ 0. The case Δ < 0 can be dealt with similarly. By moving the rectangle θF_{n} to the origin, straightforward geometric considerations lead to the following facts:

(F1)

T (L) = L^{(ϵ)} where ϵ := \frac{4θ (k^{−(n+1)(1+j)} + Δk^{−(n+1)(1+i)})}{\sqrt{1 + Δ^{2}}}

(F2) T (L) ∩ θF_{n} ⊂ F (c; l_{1}, l_{2}) where F (c; l_{1}, l_{2}) is the rectangle with the same centre c as F_{n} and of size 2l_{1} × 2l_{2} with

l_{1} := \frac{θ}{Δ} (k^{−n(1+j)} + 4k^{−(n+1)(1+j)} + Δk^{−(n+1)(1+i)})
l_{2} := θk^{−n(1+j)} .

We now estimate the right hand side of (16) by considering two cases. Throughout, let a_{i}, b_{i} denote the constants associated with the measure m_{i} and condition (A) and let

ϖ := 3 (\frac{4b_{1}b_{2}}{κ_{1} a_{1}a_{2}2^{δ_{1}+δ_{2}}} )^{1/δ_{1}} .

Case (i): Δ ≥ ϖk^{−n(1+j)}/k^{−n(1+i)}. In view of (F2) above, we trivially have that

m(θF_{n} ∩ T (L)) ≤ m(F (c; l_{1}, l_{2})) ≤ b_{1} b_{2} l_{1}^{δ_{1}} l_{2}^{δ_{2}} .

It follows that

\frac{m(T (L) ∩ θF_{n})}{m(2θF_{n+1})} ≤ \frac{b_{1}b_{2}l_{1}^{δ_{1}}l_{2}^{δ_{2}}}{a_{1}a_{2}(2θ)^{δ_{1}+δ_{2}} k^{−(n+1)(1+j)δ_{1}} k^{−(n+1)(1+i)δ_{2}}}

≤ \frac{b_{1}b_{2}}{a_{1}a_{2}2^{δ_{1}+δ_{2}}} (\frac{1}{ϖ} + \frac{1}{ϖk^{1+j}} + \frac{1}{k^{1+i}} )^{δ_{1}} k^{(1+j)δ_{1}} k^{(1+i)δ_{2}}

≤ \frac{b_{1}b_{2}}{a_{1}a_{2}2^{δ_{1}+δ_{2}}} (\frac{3}{ϖ} )^{δ_{1}} k^{(1+j)δ_{1}} k^{(1+i)δ_{2}}

= \frac{κ_{1}}{4} k^{(1+j)δ_{1}}k^{(1+i)δ_{2}} .

Case (ii): 0 ≤ Δ < ϖk^{−n(1+j)}/k^{−n(1+i)}. By the covering lemma of §3, there exists a collection B_{n} of disjoint balls B_{n} with centres in θF_{n} ∩ Ω and radii θk^{−n(1+j)} such that

θF_{n} ∩ Ω ⊂ ⋃_{B_{n}∈B_{n}} 3B_{n} .

Since i < j, it is easily verified that the disjoint collection B_{n} is contained in 2θF_{n} and thus #B_{n} ≤ m(2θF_{n})/m(B_{n}). It follows that

m(θF_{n} ∩ T (L)) ≤ m (∪_{B_{n}∈B_{n}}3B_{n} ∩ T (L))
≤ #B_{n} m(3B_{n} ∩ T (L))
≤ \frac{m(2θF_{n})}{m(B_{n})} m (3B_{n} ∩ L^{(ϵ)})  by (F1) above
≤ m(2θF_{n}) \frac{m(3B_{n})}{m(B_{n})} (\frac{ϵ}{3θk^{−n(i+j)}} )^{α} m is absolutely α-decaying.

Now notice that

\frac{ϵ}{3θk^{−n(i+j)}} ≤ \frac{4}{3} (k^{−(1+j)} + ϖk^{−(1+i)}) .

Hence, for k sufficiently large we have that

\frac{m(T (L) ∩ θF_{n})}{m(2θF_{n+1})} ≤ \frac{κ_{1}}{4} k^{(1+j)δ_{1}}k^{(1+i)δ_{2}} .

On combining the above two cases, we have that

l.h.s. of (6) ≤ \frac{m(T (L) ∩ θF_{n})}{m(2θF_{n+1})} ≤ \frac{κ_{1}}{4} k^{(1+j)δ_{1}}k^{(1+i)δ_{2}} = \frac{1}{4} × l.h.s. of (14) .

Hence (6) is satisfied and Theorem 3 implies the desired result.

□

The argument used to establish Theorem 11 can be adapted in the obvious manner to prove a slightly more general result.

Theorem 13 For 1 ≤ j ≤ N, let Ω_{j} be a compact subset of ℝ^{d_{j}} which supports an absolutely α_{j}-decaying measure m_{j} satisfying condition (A) with exponent δ_{j}. Let Ω denote the product set Ω_{1} × … × Ω_{N}. Then, for any N-tuple (i_{1}, …, i_{N}) with i_{j} ≥ 0 and ∑_{j=1}^{N} d_{j} i_{j} = 1,

dim Bad_{Ω}( \underbrace{i_{1}, …, i_{1}}_{d_{1} times} ; \underbrace{i_{2}, …, i_{2}}_{d_{2} times} ; …; \underbrace{i_{N} , …, i_{N}}_{d_{N} times} ) = dim Ω = ∑_{j=1}^{N} δ_{j} .

The following is a simple consequence of Theorem 9 and Theorem 13.

Corollary 14 For 1 ≤ j ≤ N, let K_{j} be the attractor of a finite irreducible family of contracting self similarity maps of ℝ^{d_{j}} satisfying the open set condition. Let m_{j} be the restriction of H^{δ_{j}} to K_{j} where δ_{j} = dim K_{j}. Let K denote the ‘product attractor’ K_{1} × … × K_{N}. Then, for any N-tuple (i_{1}, …, i_{N}) with i_{j} ≥ 0 and ∑_{j=1}^{N} d_{j} i_{j} = 1,

dim(K ∩ Bad( \underbrace{i_{1}, …, i_{1}}_{d_{1} times} ; \underbrace{i_{2}, …, i_{2}}_{d_{2} times} ; …; \underbrace{i_{N} , …, i_{N}}_{d_{N} times} )) = dim K .

As an application of Corollary 14 we obtain the following statement which to some extent is more illuminating – even this special case appears to be new.

Corollary 15 Let V ⊂ ℝ^{2} be the von Koch curve and K ⊂ ℝ be the middle third Cantor set. Then, for any positive i and j with 2 i + j = 1

dim ((V × K) ∩ Bad(i, i, j)) = dim(V × K) = \frac{log 8}{log 3} .

5.1.1 Remarks related to Schmidt’s conjecture

In §2.3, we mentioned the result that dim(Bad(i, j)∩Bad(1, 0)∩Bad(0, 1)) = 2. This can easily be obtained via Theorem 11. To see this, first of all notice that Bad × Bad = Bad(1, 0) ∩ Bad(0, 1). For M ≥ 2, let F_{M} := {x ∈ [0, 1] : x := [a_{1}, a_{2}, …] with a_{i} ≤ M for all i}. Thus F_{M} is the set of real numbers in the unit interval with partial quotients bounded above by M. By definition F_{M} is a compact subset of Bad and moreover it is well known that F_{M} supports a measure m_{M} which satisfies condition (A) with exponent δ_{M} with δ_{M} → 1 as M → ∞. Now let Ω := F_{M} × F_{M}, then Theorem 11 implies that

dim(Bad(i, j) ∩ Bad(1, 0) ∩ Bad(0, 1)) ≥ dim(Bad_{Ω}(i, j)) = 2δ_{M} .

On letting M → ∞, we obtain that dim(Bad(i, j)∩Bad(1, 0)∩Bad(0, 1)) ≥ 2. The complementary upper bound result is trivial since the set in question is a subset of ℝ^{2}.

Recall, that Schmidt’s conjecture [22] states that Bad(i, j)∩Bad(i′, j′) ≠ ∅. In fact, Schmidt stated this conjecture in the simpler situation when i = j′ = 1/3 and i′ = j = 2/3. Even this specific and symmetric case is unsolved. In order to illustrate a possible approach towards the conjecture via the results of this paper we consider the special case of Bad(i, j) ∩ Bad(1/2, 1/2). Suppose for the moment that we could find a compact set Ω ⊆ Bad(i, j) with a measure m satisfying condition (A) for some δ > 1. Let ρ(r) = r^{−3/2}. Using Lemma 7 together with the ‘triangle’ argument or equivalently the Simplex Lemma of §2.3, we may construct collections C(θB_{n}) as in the statement of Theorem 1. The condition that δ > 1 is used to ensure (2). This leads to the following enticing statement:

If there exists a compact subset Ω of Bad(i, j) which supports a measure m satisfying condition (A) with exponent δ > 1, then

dim(Bad(i, j) ∩ Bad(1/2, 1/2)) ≥ δ .

Clearly, this would imply that Bad(i, j) ∩ Bad(1/2, 1/2) ≠ ∅. Regarding the above statement, it is not particularly difficult to prove the existence of a compact subset Ω supporting a measure m satisfying condition (A) with δ < 1. However, from this we are not able to deduce that dim(Bad(i, j) ∩ Bad(1/2, 1/2)) ≥ δ or even that Bad(i, j) ∩ Bad(1/2, 1/2) ≠ ∅.

5.2 Rational Maps

In this section we consider the ‘badly approximable’ analogue of the ‘shrinking target’ problem introduced in [8] for expanding rational maps. Let T be an expanding rational map (degree ≥ 2) of the Riemann sphere \overline{ℂ} = ℂ ∪ \{∞\} and J(T) be its Julia set. For any z_{0} ∈ J(T) consider the set

Bad_{z_{0}}(J) := {z ∈ J(T) : ∃ c(z) > 0 such that T^{n}(z) ∉ B (z_{0}, c(z)) for any n ∈ ℕ} .

Clearly, the forward orbit of points in Bad_{z_{0}}(J) are not dense in J(T). Now let m be Sullivan measure and δ = dim J(T). Thus m is a non-atomic, δconformal probability measure supported on J(T) and since T is expanding it satisfies condition (A). Moreover, m is equivalent to δ-dimensional Hausdorff measure H^{δ} – see [8,9] for the details. In view of the ‘Khintchine type’ result for expanding rational maps (see, for example [3, §8.4]) it is easily verified that H^{δ}(Bad_{z_{0}}(J)) = 0 = m(Bad_{z_{0}}(J)). Nevertheless, the set Bad_{z_{0}}(J) is large in that it is of maximal dimension.

Theorem 16

dim Bad_{z_{0}}(J) = δ .

This result is not new and has been established by numerous people (see e.g. [2]). However, we give a short proof which indicates the versatility and generality of our framework and results.

Proof of Theorem 16. In view of the bounded distortion property for expanding maps [8, Proposition 1], we can rewrite Bad_{z_{0}}(J) in terms of points in the Julia set which ‘stay clear’ of balls centred around the backward orbit of the selected point z_{0}:

Bad_{z_{0}}(J) ≡ {z ∈ J(T) : ∃ c(z) > 0 such that z ∉ B (y, c(z)|(T^{n})′(y)|^{−1}) for any (y, n) ∈ I} ,

where I := {(y, n) : n ∈ ℕ with T^{n}(y) = z_{0}}. Also, since T is expanding, J(T) can be thought of as a compact metric space with the usual metric on ℂ. It is now clear that Bad_{z_{0}}(J) can be expressed in the form Bad^{∗}(R, β, ρ) with ρ(r) := r^{−1} and

X = Ω := J(T) , J := I , α := (y, n) ∈ I , β_{α} := |(T^{n})′(y)| , R_{α} := y .

With reference to Theorem 1, Sullivan measure m and the function ρ satisfy condition (A) and (B) respectively. To deduce Theorem 16 from Theorem 1 we need to establish the existence of the disjoint collection C(θB_{n}) of balls 2θB_{n+1} where B_{n} is an arbitrary ball of radius k^{−n} with centre in Ω. In view of Lemma 7, for k sufficiently large, there exists a disjoint collection C(θB_{n}) such that

#C(θB_{n}) ≥ κ_{1} k^{δ} ;  (17)

i.e. (1) of Theorem 1 holds. We now verify that (2) is satisfied for any such collection. First we recall a key result which is the second part of the statement of Lemma 8 in [9]. For ease of reference we keep the same notation and numbering of constants as in [9].

Constant Multiplicity: For X ∈ ℝ^{+}, let P (X) denote the set of pairs (y, n) ∈ I such that f_{n}(y) − C_{8} ≤ X ≤ f_{n+1}(y) + C_{8}, where f_{n}(y) := log |(T^{n})′(y)|. Let z ∈ J(T). Then there are no more than C_{9} pairs (y, n) ∈ P (X) such that z ∈ B (y, C_{10} |(T^{n})′(y)|^{−1}).

We are now in the position to verify (2) of Theorem 1. By definition J(n+1) := {(y, m) ∈ I : k^{n−1} ≤ |(T^{m})′(y)| < k^{n}} and let θ := C_{10}k^{−1}. It follows that

l.h.s. of (2) ≤ #{y ∈ θB_{n} : (y, m) ∈ J(n + 1)}
≤ #{y ∈ B(c, C_{10}|(T^{m})′(y)|^{−1}) : (y, m) ∈ J(n + 1)} , (18)

where c is the centre of θB_{n}. Without loss of generality, assume that |T′(z_{0})| > 1. Otherwise, since T is expanding we simply work with some higher iterate T^{q} of T for which |(T^{q})′(z_{0})| > 1. Then, the chain rule together with the above ‘constant multiplicity’ fact implies that the r.h.s. of (18) is ≪ C_{9} log k. Hence, for k sufficiently large

l.h.s. of (2) ≤ \frac{1}{2} × r.h.s. of (17) .

Thus, (2) is easily satisfied and Theorem 1 implies Theorem 16. □

Remark: It is worth mentioning that our framework also yields (just as easily) the analogue of Theorem 16 within the Kleinian group setup. Briefly, let G be either a geometrically finite Kleinian group of the first kind or a convex co-compact group and let Λ(G) denote its limit set. For these groups, Patterson measure supported on Λ(G) satisfies condition (A) and plays the role of Sullivan measure. Then, it is not difficult to obtain the Kleinian group analogue of Theorem 16 via Theorem 1; i.e. the set of ‘badly approximable’ limit points is of full dimension – dim Λ(G).

5.3 Complex numbers

In this section we consider the badly approximable analogue of Bad(i_{1}, …, i_{N}) in ℂ^{N}. Let N ∈ ℕ and i_{1}, …, i_{N} ≥ 0 such that i_{1} + ⋯ + i_{N} = 1. Now define the set Bad_{ℂ}(i_{1}, …, i_{N}) to consist of z := (z_{1}, …, z_{N}) ∈ ℂ^{N} for which there exists a constant c(z) > 0 such that for any q, p_{1}, …, p_{N} ∈ ℤ[i], q ≠ 0,

max{|qz_{1} − p_{1}|^{1/i_{1}}, …, |qz_{N} − p_{N}|^{1/i_{N}} } ≥ c(z)|q|^{−1} .

In the case i_{1} = ⋯ = i_{N} = 1/N, the corresponding set will be denoted by Bad_{ℂ}(N). Notice, that the role of the rationals in the real setup is replaced by ratios of Gaussian integers in the complex setup. We shall refer to the latter as Gaussian points.

The Hausdorff dimension of the set Bad_{ℂ}(N) has been studied in the past by various people using Kleinian groups [4], Riemannian geometry [7] and Schmidt’s (α, β)-games [5]. Theorem 1 of this paper will also give the Hausdorff dimension of this set. In fact, our general framework enables us to find the dimension of Bad_{ℂ}(i_{1}, …, i_{N}) intersected with direct products of sets supporting measures satisfying condition (A). As a consequence, the previously known results are extended to the ‘rectangular’ or ‘weighted’ form of simultaneous approximation in ℂ^{N}. The following statement is the ‘complex’ analogue of Theorem 11.

Theorem 17 For 1 ≤ j ≤ N, let Ω_{j} be a compact subset of ℂ which supports a measure m_{j} satisfying condition (A) with exponent δ_{j}. Let Ω denote the product set Ω_{1} × … × Ω_{N}. Then, for any N-tuple (i_{1}, …, i_{N}) with i_{j} ≥ 0 and ∑_{j=1}^{N} i_{j} = 1,

dim(Bad_{ℂ}(i_{1}, …, i_{N}) ∩ Ω ) = dim Ω .

The following complex notion of absolutely decaying measures will be useful in proving the above theorem. Let Ω be a compact subset of ℂ^{N} which supports a non-atomic, finite measure m. Let L denote a generic (N − 1)-dimensional complex hyperplane of ℂ^{N} and let L^{(ϵ)} denote its ϵ-neighborhood. We say that m is absolutely α-decaying if there exist strictly positive constants C, α, r_{0} such that for any complex hyperplane L, any ϵ > 0, any z ∈ Ω and any r < r_{0},

m (B(z, r) ∩ L^{(ϵ)}) ≤ C (\frac{ϵ}{r} )^{α} m(B(z, r)) .

Note that if N = 1, so that Ω is a subset of ℂ, the complex hyperplane L is simply a point a ∈ ℂ and L^{(ϵ)} is the ball B(a, ϵ) centred at a of radius ϵ. Moreover, if the measure m satisfies condition (A) with exponent δ then m is automatically absolutely δ-decaying.

It is easy to verify that the statement of the ‘Fact’ in §5.1 regarding the product of absolutely decaying measures remains valid for the complex notion.

Proof of Theorem 17 (Sketch). As usual we restrict our attention to the case N = 2 and write Bad_{ℂ}(i, j) for Bad_{ℂ}(i_{1}, i_{2}). Assume that i ≤ j. Clearly, the set Bad_{ℂ}(i, j) ∩ Ω can be expressed in the form Bad^{∗}(R, β, ρ_{1}, ρ_{2}) with ρ_{1}(r) = r^{−(1+i)}, ρ_{2}(r) = r^{−(1+j)} and

X = (ℂ^{2}, d) , J := {((p_{1}, p_{2}), q) ∈ ℤ[i]^{2} × ℤ[i]\{0}} ,
α := ((p_{1}, p_{2}), q) ∈ J , β_{α} := |q| , R_{α} := (p_{1}/q, p_{2}, q) .

The metric d on ℂ^{2} is the maximum of the coordinate metrics; i.e. d((z_{1}, z_{2}), (z′_{1}, z′_{2})) = max{d(z_{1}, z′_{1}), d(z_{2}, z′_{2})}. Also note that the measure m := m_{1} × m_{2} is absolutely α-decaying on Ω with α := min{δ_{1}, δ_{2}}. This follows from the above discussion concerning the complex notion of absolutely decaying measures and their product.

With reference to Theorem 3, we need to establish the existence of the collection C(θF_{n}) where F_{n} is an arbitrary closed polydisc B_{n,1} × B_{n,2} with centre c in Ω. Here B_{n,1} (resp. B_{n,2}) is a closed ball in ℂ of radius k^{−n(1+i)} (resp. k^{−n(1+j)}). In view of Lemma 7, there exists a disjoint collection C(θF_{n}) of polydiscs 2θF_{n+1} ⊂ θF_{n} such that (5) of Theorem 3 is satisfied. We now verify that (6) is satisfied for any such collection by modifying the proof of Theorem 11 in the obvious manner. The only part which is not so obvious is the complex analogue of the ‘triangle’ argument of §2.3. For this suppose that θF_{n} is given and that there are at least three Gaussian points (p_{1}/q, p_{2}/q), (p′_{1}/q′, p′_{2}/q′) and (p′′_{1}/q′′, p′′_{2}/q′′) with

k^{n} ≤ |q|, |q′|, |q′′| < k^{n+1}

lying within θF_{n}. Suppose for the moment that they do not lie on a one– dimensional complex hyperplane (i.e. a complex line) L of ℂ^{2} and consider the determinant

D = det (\begin{matrix}1 & p_{1}/q & p_{2}/q \\ 1 & p′_{1}/q′ & p′_{2}/q′ \\ 1 & p′′_{1}/q′′ & p′′_{2}/q′′ \end{matrix}) ≠ 0 .

Expanding the determinant in the first column and using the fact that the ring of Gaussian integers is a unique factorization domain, we find that

|D| > \frac{1}{k^{3(n+1)}} .

On the other hand, the absolute value of D can be at most twice the diameters of the two projections θB_{n,1} and θB_{n,2} of θF_{n}. That is

|D| ≤ 2 \frac{2 θ}{k^{n(i+1)}} \frac{2 θ}{k^{n(j+1)}} = \frac{8θ^{2}}{k^{3n}} .

To see this, note that for (z_{1}, z_{2}), (z′_{1}, z′_{2}), (z′′_{1}, z′′_{2}) ∈ θF_{n}

|det (\begin{matrix}1 & z_{1} & z_{2} \\ 1 & z′_{1} & z′_{2} \\ 1 & z′′_{1} & z′′_{2} \end{matrix})| = |(z_{1} − z′_{1})(z′_{2} − z′′_{2}) + (z′_{1} − z′′_{1})(z′_{2} − z_{2})|

≤ 2 × 2θρ_{1}(k^{n}) 2θρ_{2}(k^{n}) .

Now with θ := (8k^{3})^{−1/2}, we obtain the desired contradiction. Thus, if there are two or more Gaussian points with k^{n} ≤ |q| < k^{n+1} lying within θF_{n} then they must lie on a complex line L. It now follows that

l.h.s. of (6) ≤ #{2θF_{n+1} ⊂ C(θF_{n}) : 2θF_{n+1} ∩ L ≠ ∅} .

This is the precise complex analogue of (15) and the proof can now be completed by modifying the proof of the real case (Theorem 11) in the obvious manner. We leave the details to the reader. □

It is worth mentioning that Theorem 17 can be generalized in the obvious manner to obtain the complex analogue of Theorem 13.

5.4 p-adic numbers

For a prime p, let | · |_{p} denote the p-adic absolute value and let ℚ_{p} denote the p-adic field. Furthermore, let ℤ_{p} denote the ring of p-adic integers. In this section we consider the badly approximable analogue of Bad(i_{1}, …, i_{N}) in ℤ_{p}^{N}. Let N ∈ ℕ and i_{1}, …, i_{N} ≥ 0 such that i_{1} + ⋯ + i_{N} = 1. Now define the set Bad_{ℤ_{p}}(i_{1}, …, i_{N}) to consist of x := (x_{1}, …, x_{N}) ∈ ℤ_{p}^{N} for which there exists a constant c(x) > 0 such that

max{|qx_{1} − r_{1}|_{p}^{1/(1+i_{1})}, …, |qx_{N} − r_{N}|_{p}^{1/(1+i_{N})}}
≥ c(x) max{|r_{1}|, …, |r_{N}|, |q|}^{−1} ,  (19)

for all ((r_{1}, …, r_{N}), q) ∈ ℤ^{N} × ℤ \ {0}. In the case i_{1} = ⋯ = i_{N} = 1/N, the corresponding set will be denoted by Bad_{ℤ_{p}}(N).

There are two points worth making when comparing the above set with the ‘classical’ set Bad(i_{1}, ..., i_{N}). Firstly, the r.h.s of (19) in the p-adic setup is a function of max(|r_{1}|, …, |r_{N}|, |q|) rather than simply |q|. This is due to the fact that within the p-adic setup for any x ∈ ℤ_{p}^{N} and q ∈ ℤ there exists r ∈ ℤ^{N} such that the l.h.s of (19) can be made arbitrarily small. Thus, the set of x ∈ ℤ_{p}^{N} for which l.h.s of (19) ≥ c(x) |q|^{−1} is in fact empty and there is nothing to prove. Secondly, in the p-adic setup the ‘weighting’ factor occurring on the l.h.s of (19) is 1/(1 + i_{s}) rather than 1/i_{s} (1 ≤ s ≤ N). This is due to the fact that we approximate in terms of the p-adic absolute value on the left hand side, but measure the ‘rate’ of approximation in terms of the ordinary absolute value on the right hand side. Because of the arithmetical properties of the p-adic absolute value, we generally expect the ‘rate’ of the approximation to be better (see below).

Badly approximable p-adic numbers have in the past been studied by Abercrombie [1], who showed that Bad_{ℤ_{p}}(1) has full Hausdorff dimension. In higher dimensions, the corresponding result for even the ‘symmetric’ set Bad_{ℤ_{p}}(N) is unknown. Using the framework established in this paper, we are able to prove the following complete result.

Theorem 18

dim Bad_{ℤ_{p}}(i_{1}, …, i_{N}) = N .

Proof of Theorem 18 (Sketch). As in the preceding applications, we restrict our attention to the case N = 2 and write Bad_{ℤ_{p}}(i, j) for Bad_{ℤ_{p}}(i_{1}, i_{2}). Assume that i ≤ j. Clearly the set Bad_{ℤ_{p}}(i, j) can be expressed in the form Bad^{∗}(R, β, ρ_{1}, ρ_{2}) with ρ_{1}(x) := x^{−(1+i)}, ρ_{2}(x) := x^{−(1+j)} and

X = Ω := ℤ_{p}^{2} = ℤ_{p} × ℤ_{p} , J := {((r_{1}, r_{2}), q) ∈ ℤ^{2} × ℤ \ {0}} ,
α := ((r_{1}, r_{2}), q) ∈ J , β_{α} := max{|r_{1}|, |r_{2}|, |q|} ,
R_{α} := {(x_{1}, x_{2}) ∈ ℤ_{p}^{2} : qx_{1} = r_{1}, qx_{2} = r_{2}} .

Furthermore, d = d_{1} × d_{1} where d_{1}(x, y) := |x − y|_{p} is the p-adic metric on ℚ_{p} and m := μ × μ where μ is normalized Haar measure on ℚ_{p}. Thus, μ(ℤ_{p}) = 1 and μ(B(x, p^{−t})) = p^{−t} for any t ∈ ℕ. Note that these are the only radii which make sense – if p^{−t} ≤ r < p^{−t+1}, then B(x, r) = B(x, p^{−t}).

We take a moment to verify that the set Bad(i, j) is indeed equal to the set Bad^{∗}(R, β, ρ_{1}, ρ_{2}). Fix q ∈ ℤ \ {0} and (r_{1}, r_{2}) ∈ ℤ^{2}. Associated with the pair ((r_{1}, r_{2}), q) is the resonant point R_{((r_{1},r_{2}),q)} = (R_{(r_{1},q)}, R_{(r_{2},q)}). First, note that |qx_{s} − r_{s}|_{p} = |q|_{p} d_{1}(x_{s}, R_{(r_{s},q)}) for s ∈ {1, 2}. However, |q|_{p} ≤ 1 and so clearly Bad(i, j) ⊆ Bad^{∗}(R, β, ρ_{1}, ρ_{2}). Conversely, let x ∈ Bad^{∗}(R, β, ρ_{1}, ρ_{2}). We show that (19) is satisfied for r and q. If (q, p) = 1, then |q|_{p} = 1 and the inequality is immediate. If p^{t}|q for some t ∈ ℕ, but either (r_{1}, p) = 1 or (r_{2}, p) = 1, the inequality is also satisfied. To see this, suppose that (r_{1}, p) = 1 and express −r_{1} and qx_{1} as power series in p. Clearly, the lowest exponent of p in the expansion of qx_{1} it at least t, whereas the expansion of −r_{1} has a term with exponent zero. Hence the sum of the two must have a term of exponent zero, and so |qx_{1} − r_{1}|_{p} = 1 and we are done. In the remaining case, when p divides q, r_{1} and r_{2}, we simply factor out the highest possible power of p in the left hand side of (19) and the problem reduces to one of the previous cases. Thus, Bad^{∗}(R, β, ρ_{1}, ρ_{2}) ⊆ Bad(i, j).

With reference to Theorem 3, the functions ρ_{1}, ρ_{2} satisfy condition (B*) and the measures m_{1} := μ and m_{2} := μ satisfy condition (A) with δ_{1} = δ_{2} = 1. We need to establish the existence of the collection C(θF_{n}) where F_{n} is an arbitrary closed rectangle of size 2k^{−n(1+i)} ×2k^{−n(1+j)}. Here, we take k = p^{s} and θ = p^{−t} for some s, t ∈ ℕ which will be chosen sufficiently large later on. In view of Lemma 7, there exists a disjoint collection C(θF_{n}) of rectangles 2θF_{n+1} ⊂ θF_{n} such that (5) of Theorem 3 is satisfied. We now verify that (6) is satisfied for any such collection. This follows by modifying the ‘triangle’ argument of §2.3 to the p-adic setting. So, let us assume that we have three resonant points (which by definition are rational points) (r_{1}/q, r_{2}/q), (r′_{1}/q′, r′_{2}/q′) and (r′′_{1}/q′′, r′′_{2}/q′′) lying in some rectangle θF_{n} with

k^{n} ≤ max_{1≤s,t,u≤2}{|r_{s}|, |r′_{t}|, |r′′_{u}|, |q|} < k^{n+1}.  (20)

Suppose that they do not lie on a line. Then, they span a p-adic triangle Δ. By results in Lutz [14, Chapter I, §4], the Haar measure m of Δ is comparable to

|det (\begin{matrix}1 & r_{1}/q & r_{2}/q \\ 1 & r′_{1}/q′ & r′_{2}/q′ \\ 1 & r′′_{1}/q′′ & r′′_{2}/q′′ \end{matrix})|_{p} ≠ 0.

The determinant is a rational number with denominator qq′q′′. As these are integers, the p-adic absolute value is ≤ 1. Hence, the absolute value of the determinant is bounded below by the p-adic absolute value of the enumerator:

N = r_{1}r′_{2}q′′ − r_{2}r′_{1}q′′ − r_{1}q′r′′_{2} + r_{2}q′r′′_{1} + qr′_{1}r′′_{2} − qr′_{2}r′′_{1}.

This is an integer. In view of (20), we have that

|N| < 6k^{3n+3}.

We may assume without loss of generality that N > 0. Clearly, the p-adic valuation v_{p}(N) (i.e. the number of times p divides N) satisfies

v_{p}(N) < log_{p}(6k^{3n+3}).

But |N|_{p} = p^{−v_{p}(N)} so that

|N|_{p} > p^{− log_{p}(6k^{3n+3})} = 1/(6k^{3n+3}) .

Hence, there is a constant C > 0 such that m(Δ) > C/(6k^{3n+3}). However, μ(θF_{n}) ≤ θ^{2}k^{−3n} and on choosing θ^{2} := p^{−2t} < C/(6k^{3}) we obtain the desired contradiction; i.e. by choosing t sufficiently large. Thus, it there are two or more resonant points satisfying (20) lying within θF_{n} then they must lie on a p-adic line L. It now follows that

l.h.s. of (6) ≤ #{2θF_{n+1} ⊂ C(θF_{n}) : 2θF_{n+1} ∩ L ≠ ∅} .

A simple geometric argument, analogous to that employed in §2.3, ensures that the line L can not pass through more than C′ k^{j+1} of the 2θF_{n+1} rectangles. Here C′ > 0 is a constant independent of k. On choosing k := p^{s} sufficiently large (i.e. s large enough), we ensure that C′ k^{1+j} < κ_{1} k^{3} which establishes (6) and thereby completes the proof of the theorem. □

Under suitable assumptions on subsets Ω_{i} of ℤ_{p} with measures satisfying condition (A), we can also obtain the p-adic analogues of Theorems 11 and 13. Of course, to achieve this, one also needs to assume the natural p-adic analogue of a measure being absolutely α-decaying.

5.5 Formal power series

Apart from the p-adics, badly approximable elements have been extensively studied over another locally compact ultra-metric field. Let 𝔽 be the finite field with h elements. Thus, h = p^{r} for some prime p and r ∈ ℕ. Now define

𝔽((X^{−1})) := { ∑_{i=−n}^{∞} a_{−i}X^{−i} : n ∈ ℤ, a_{i} ∈ 𝔽, a_{n} ≠ 0} ∪ {0} ,

with an absolute value

∥ ∑_{i=−n}^{∞} a_{−i}X^{−i}∥ := h^{n}, ∥0∥ := 0 .

Under ordinary addition and multiplication, this is a locally compact field. The closed unit ball I = {x ∈ 𝔽((X^{−1})) : ∥x∥ ≤ 1} is a compact subspace of this space.

In this section we consider the badly approximable analogue of Bad(i_{1}, …, i_{N}) in I^{N}. Let N ∈ ℕ and i_{1}, …, i_{N} ≥ 0 such that i_{1} + ⋯ + i_{N} = 1. Now define the set Bad_{𝔽((X^{−1}))}(i_{1}, …, i_{N}) to consist of x := (x_{1}, …, x_{N}) ∈ 𝔽((X^{−1}))^{N} for which there exists a constant c(x) > 0 such that

max{∥qx_{1} − p_{1}∥^{1/i_{1}}, …, ∥qx_{N} − p_{N}∥^{1/i_{N}} } ≥ c(x) ∥q∥^{−1}

for all q, p_{1}, …, p_{N} ∈ 𝔽 [X] (q ≠ 0). Note that in this setup, the polynomial ring 𝔽 [X] plays the role of the integers. When i_{1} = … = i_{N} = 1/N, the corresponding set will be denoted by Bad_{𝔽((X^{−1}))}(N). Niederreiter and Vielhaber [17] have shown that the set Bad_{𝔽((X^{−1}))}(1) has full dimension. Using the framework established in this paper, we are able to obtain the complete result for the ‘weighted’ simultaneous set.

Theorem 19

dim Bad_{𝔽((X^{−1}))}(i_{1}, …, i_{N}) = N .

Proof of Theorem 19 (Sketch). As usual, we restrict our attention to the case N = 2 and write Bad_{𝔽((X^{−1}))}(i, j) for Bad_{𝔽((X^{−1}))}(i_{1}, i_{2}). In view of the geometrical nature of our approach and the similarities between this situation and the preceding ones (in particular the p-adic case), we only outline the modifications needed to deal with the present situation in the briefest sense. The field 𝔽((X^{−1})) supports a Haar measure m satisfying m(B(c, h^{−t})) = h^{−t} for all t ∈ ℤ. As was the case in the p-adics, these are the only balls for which a calculation is needed. Let I denote the unit ball in this space. We set X_{1} = X_{2} = 𝔽((X^{−1})), Ω_{1} = Ω_{2} = I with the metrics induced by the absolute value and Haar measure defined above. We let J = {((p_{1}, p_{2}), q) ∈ 𝔽[X]^{2} × 𝔽[X] \ {0}} and for any ((p_{1}, p_{2}), q) ∈ J, we let β_{((p_{1},p_{2}),q)} = ∥q∥. The resonant sets R_{((p_{1},p_{2}),q)} = (p_{1}/q, p_{2}/q). Finally, define functions ρ_{1}(x) = x^{−(i+1)} and ρ_{2}(x) = x^{−(j+1)}. Clearly, the conditions of Theorem 3 are satisfied and the set Bad_{𝔽((X^{−1}))}(i, j) ∩ I^{2} = Bad^{∗}(R, β, ρ_{1}, ρ_{j}).

We establish the collection C(θF_{n}) by Lemma 7. The triangle argument works in this setting by results of Mahler [15] to calculate the measures of the sets involved. Note that in this case, the lower bound on the denominator is the important feature in the argument, so the proof differs from the p-adic case in this respect. Finally, maximal number of rectangles in C(2θF_{n+1}) with nontrivial intersection with the resulting ‘line’ is estimated by arguments as in the p-adic case. □

As in the p-adic setup, under appropriate assumptions we can also obtain the formal power series analogues of Theorems 11 and 13. We have chosen to restrict ourselves to the simpler situation, as this already yields new results and illustrates the versatility of our framework.

Acknowledgments: SV would like to thank Dmitry Kleinbock and Barak Weiss for the many e-mail conversations regarding the development of our separate approaches. Also, he’d like to take this opportunity to say ‘marra peeta’ to Iona and Ayesha.

References

[1] A.G. Abercrombie : Badly approximable p-adic integers, Proc. Indian Acad. Sci. Math. Sci., 105(2) (1995) 123–134.

[2] A.G. Abercrombie and R. Nair : An exceptional set in the ergodic theory of rational maps of the Riemann sphere, Erg. Th. & Dyn. Sys., 17(2), (1997), 253–267.

[3] V. Beresnevich, H. Dickinson and S.L. Velani : Measure Theoretic Laws for limsup sets, Mem. Amer. Math. Soc. To appear. Pre-print on http://www.

arxiv.org/abs/math.NT/0401118.

[4] C.J. Bishop and P.W. Jones : Hausdorff dimension and Kleinian groups, Acta Math., 179(1) (1997) 1–39.

[5] M.M. Dodson and S. Kristensen : Hausdorff dimension and Diophantine approximation, in Fractal geometry and applications: a jubilee of Benoît Mandelbrot. Part 1, 305–347, Proc. Sympos. Pure Math., Part 1, Amer. Math. Soc., Providence, RI, 2004.

[6] K. Falconer : Fractal Geometry: Mathematical Foundations and Applications. John Wiley & Sons, (1990).

[7] J.L. Fernández and M.V. Melián : Bounded geodesics of Riemann surfaces and hyperbolic manifolds, Trans. Amer. Math. Soc., 347(9) (1995) 3533–3549.

[8] R. Hill and S.L. Velani : Ergodic theory of shrinking targets. Invent. Math., 119 (1995) 175-198.

[9] R. Hill and S.L. Velani : Metric Diophantine approximation in Julia sets of expanding rational maps, Inst. Hautes Études Sci. Publ. Math., 85 (1997) 193-216.

[10] I. Jarník : Zur metrischen Theorie der diophantischen Appoximationen. Prace Mat.-Fiz., 36 (1928–29) 91–106.

[11] D. Kleinbock, E. Lindenstrauss and B. Weiss : On fractal measures and Diophantine approximation. Selecta Math. To appear. Pre-print on http:

//people.brandeis.edu/^{∼}kleinboc/Pub/friendly.pdf.

[12] D. Kleinbock and B. Weiss : Badly approximable vectors on fractals. Israel

J. Math. To appear. Pre-print on http://people.brandeis.edu/^{∼}kleinboc/
Pub/Pub/bad.pdf.

[13] S. Kristensen and S.L. Velani : Diophantine approximation and badly approximable sets II, In preparation.

[14] E. Lutz : Sur les approximations diophantiennes linéaires P -adiques, Actualités Sci. Ind., no. 1224. Hermann & Cie, Paris, 1955.

[15] K. Mahler : An analogue to Minkowski’s geometry of numbers in a field of series, Ann. of Math. (2) 42 (1941) 488–522.

[16] P. Mattila : Geometry of Sets and Measures in Euclidean Spaces. Cambridge studies in advanced mathematics 44, C.U.P., (1995).

[17] H. Niederreiter and M. Vielhaber : Linear complexity profiles: Hausdorff dimensions for almost perfect profiles and measures for general profiles, J. Complexity (3) 13 (1997) 353–383.

[18] A.D. Pollington and S.L. Velani : On simultaneously badly approximable pairs, J. London Math. Soc., 66 (2002) 29–40.

[19] A.D. Pollington and S.L. Velani : Metric Diophantine approximation and ‘absolutely friendly’ measures, Selecta Math. To appear. Pre-print on http:

//www.arxiv.org/abs/math.NT/0401149.

[20] W.M. Schmidt : On badly approximable numbers and certain games, Trans. Amer. Math. Soc., 123 (1966), 178–199.

[21] W.M. Schmidt : Badly approximable systems of linear forms, J. Number Theory, 1 (1969), 139–154.

[22] W.M. Schmidt : Open problems in Diophantine approximation, Approximations diophantiennes et nombres transcendants (Luminy 1982), Progress in Mathematics, Birkhäuser, (1983)

[23] M. Urbański : Diophantine approximation of self-conformal measures, J. Number Theory. To appear. Pre-print on http://www.math.unt.edu/

∼urbanski/papers/kl050404.ps.

[24] M. Urbański : Diophantine approximation of conformal measures of onedimensional iterated function systems, Compositio Math.. To appear. Pre-print

on http://www.math.unt.edu/^{∼}urbanski/papers/bw112503.ps.