Metric Diophantine approximation and ‘absolutely friendly’ measures

Andrew Pollington ^{∗}

B.Y.U., UTAH

Sanju Velani^{†}

YORK

For Ayesha and Iona on N^{o} 2

Abstract

Let W (ψ) denote the set of ψ-well approximable points in ℝ^{d} and let K be a compact subset of ℝ^{d} which supports a measure μ. In this short note, we show that if μ is an ‘absolutely friendly’ measure and a certain μ–volume sum converges then μ(W (ψ) ∩ K) = 0. The result obtained is in some sense analogous to the convergence part of Khintchines classical theorem in the theory of metric Diophantine approximation. The class of absolutely friendly measures is a subclass of the friendly measures introduced in [2] and includes measures supported on self similar sets satisfying the open set condition. We also obtain an upper bound result for the Hausdorff dimension of W (ψ) ∩ K.

1 Introduction

1.1 The problem and results

The classical result of Dirichlet in the theory of Diophantine approximation states that for any point x = (x_{1}, …, x_{d}) ∈ R^{d}, there exist infinitely many (p, q) ∈ Z^{d} × N such that

max_{1≤i≤d} |x_{i} − p_{i}/q| ≤ q^{−(d+1)/d} .

Given a real, positive decreasing function ψ : R^{+} → R^{+}, a point x ∈ R^{d} is said to be ψ–well approximable if the above inequality remains valid with the right hand side replaced with ψ(q). We will denote by W (ψ) the set of all such points; that is

W (ψ) := {x ∈ R^{d} : max_{1≤i≤d} |x_{i} − p_{i}/q| ≤ ψ(q) for infinitely many (p, q) ∈ Z^{d} × N} .

A straightforward application of the Borel-Cantelli lemma from probability theory yields the following statement.

∗This work was supported in part by the National Science Foundation.

†Royal Society University Research Fellow.

Lemma 1

|W (ψ)|_{d} = 0 if  ∑_{r=1}^{∞}(r ψ(r))^{d} < ∞ .

Thus, if the above sum converges then almost every (with respect to d-dimensional Lebesgue measure) point x ∈ R^{d} is not ψ–well approximable. For τ ≥ 0, consider the function ψ_{τ} : r → r^{−τ} and write W (τ) for W (ψ_{τ}). In view of Dirichlet’s result, W (τ) = R^{d} for τ ≤ (d+1)/d . However, in view of the above lemma we have that |W (τ)|_{d} = 0 for τ >(d+1)/d .

Now, let K be a compact subset of R^{d} which supports a non-atomic, finite measure μ and let

W_{K}(ψ) := K ∩ W (ψ) .

In short, the problem is to determine conditions on μ and ψ under which μ(W_{K}(ψ)) = 0; i.e. μ-almost every point x ∈ R^{d} is not ψ–well approximable. Note that μ(W_{K}(ψ)) = μ(W (ψ)) since μ is supported on K. For the motivation behind the problem we refer the reader to [2, 3, 4].

In [2], Kleinbock, Lindenstrauss and Weiss introduce the notion of a ‘friendly’ measure and show that if μ is friendly then μ(W_{K}(τ)) = 0 for τ > (d+1)/d ^{1}. They also show that the class of friendly measures include (i) volume measures on nondegenerate manifolds and (ii) measures supported on self similar sets satisfying the open set condition. In full generality, the definition of friendly is rather technical and will not be reproduce here – see §2 of [2].

Our aim is to obtain a statement more in line with Lemma 1 which also implies that μ(W_{K}(τ)) = 0 for τ > (d+1)/d . To achieve this we impose conditions on μ which are stronger than those of friendly. Nevertheless, measures supported on self similar sets satisfying the open set condition are still included – see §1.2. Unfortunately, volume measures on non-degenerate manifolds and not included.

Let B(x, r) be a ball in R^{d} with centre x and radius r. The measure μ is said to be doubling if there exist strictly positive constants D and r_{0} such that

μ(B(x, 2r)) ≤ D μ(B(x, r)) ∀ x ∈ K ∀ r < r_{0} .

The following notion of ‘absolutely decaying’ is essentially taken from [2]. Let L denote a generic (d − 1)–dimensional hyperplane of R^{d} and let L^{(ϵ)} denote its ϵ-neighborhood. We say that μ is absolutely α-decaying if there exist strictly positive constants C, α, r_{0} such that for any hyperplane L and any ϵ > 0

μ (B(x, r) ∩ L^{(ϵ)}) ≤ C (\frac{ϵ}{r} )^{α} μ(B(x, r)) ∀ x ∈ K ∀ r < r_{0} .

In the case d = 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.

1They actually prove their result in the multiplicative framework.

Remark. Let B(a, r) be a ball in R^{d}. A straightforward geometric argument shows that if μ is absolutely α-decaying, then for any ϵ < 1/4

μ (B(a, ϵr)) ≤ C 2^{α} ϵ^{α} μ(B(a, r)) ∀ a ∈ R^{d} ∀ r < r_{0} .  (1)

This essentially corresponds to the condition on μ imposed in [3, 4]. Note that in the case d = 1, condition (1) is equivalent to absolutely α-decay.

Definition A measure μ is said to be absolutely α–friendly if it is doubling and absolutely α-decaying.

We prove the following analogue of Lemma 1.

Theorem 1 Let K be a compact subset of R^{d} equipped with an absolutely α–friendly measure μ. Then

μ(W_{K}(ψ)) = 0 if  ∑_{r=1}^{∞} r^{α \frac{d+1}{d} −1}ψ(r)^{α} < ∞ .

Remark. In the case when d = 1, it is possible to remove the condition that μ is doubling from the definition of absolutely α–friendly; i.e. all that is required is that μ is absolutely α–decaying – see §4.2.

The above theorem should be compared with Theorems 9 and 11 of [4]. Note that in the case that ψ_{τ} : r → r^{−τ} and τ > \frac{d+1}{d},

∑_{r=1}^{∞} r^{α \frac{d+1}{d} −1}ψ_{τ} (r)^{α} := ∑_{r=1}^{∞} r^{−1−α(τ− \frac{d+1}{d})} < ∞

and so Theorem 1 implies that μ(W_{K}(τ)) = 0 whenever μ is absolutely α–friendly. More to the point, consider the function ψ : r → r^{− \frac{d+1}{d}} (log r)^{−β} where β > 1/α. Then

∑_{r=1}^{∞} r^{α \frac{d+1}{d} −1}ψ(r)^{α} := ∑_{r=1}^{∞} r^{−1}(log r)^{−αβ} < ∞ ,

and Theorem 1 implies that μ(W_{K}(ψ)) = 0 whenever μ is absolutely α–friendly.

It will be evident from the proof that all that is actually required in establishing the theorem is that the doubling and absolutely α-decaying inequalities are satisfied at μ–almost every point in K. Also the relevance of hyperplanes in the definition of absolutely α-decaying will become crystal clear from our proof of the theorem. Essentially, on the real line R an interval I_{n} of length 1/4n^{2} can contain at most one rational p/q with n ≤ q < 2n. This follows from the trivial observation that if n ≤ q, q′ < 2n then |p/q − p′/q′| ≥ 1/qq′ > 1/(2n)^{2} and this is greater than the length of I_{n}. The higher dimension analogue of this is the following. Let B_{n} be a ball in R^{d} of radius c/n^{(d+1)/d} where c is a sufficiently small constant dependent only on d. Then any rational points p/q lying within B_{n} with n ≤ q < 2n must lie on a single (d−1)–dimensional hyperplane L. This is the key observation on which the proofs of Theorems 1 and 2 hinge.

We now turn our attention to determining an upper bound for dim W_{K}(ψ) – the Hausdorff dimension of W_{K}(ψ). For s ≥ 0, let H^{s} denote the s–dimensional Hausdorff measure - see §2.

Theorem 2 Let K be a compact subset of R^{d} equipped with an absolutely α–friendly measure μ. Furthermore, suppose there exist positive constants a, b, δ and r_{o} such that

a r^{δ} ≤ μ(B(x, r)) ≤ b r^{δ} ∀ x ∈ K ∀ r < r_{0} .  (2)

Then, for s ≤ δ

H^{s}(W_{K}(ψ)) = 0 if  ∑_{r=1}^{∞} r^{α \frac{d+1}{d} −1}ψ(r)^{α+s−δ} < ∞ .

Remark 1: Note that (2) imposed on μ trivially implies that μ is doubling. Furthermore, if δ > d−1 then (2) together with a straightforward geometric argument implies that μ is absolutely α–decaying with α := δ − (d − 1) > 0. Thus, if δ >d−1 the hypothesis that μ is absolutely α–friendly is redundant from the statement of Theorem 2.

Remark 2: If K supports a measure μ satisfying (2), then dim K = δ and moreover that 0 < H^{δ}(K) < ∞ – see [1] for the details. Now, since W_{K}(ψ) is a subset of K we have that dim W_{K}(ψ) ≤ δ and so H^{s}(W_{K}(ψ)) = 0 for any s > δ. Thus, the condition s ≤ δ in the statement of the theorem can be assumed without any loss of generality.

Given a real, positive decreasing function ψ, the lower order λ_{ψ} of 1/ψ is defined by

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

and indicates the growth of the function 1/ψ ‘near’ infinity. Note that λ_{ψ} is nonnegative since ψ is a decreasing function. A simple consequence of Theorem 2 is the following statement.

Corollary 1 Let K be a compact subset of R^{d} equipped with an absolutely α–friendly measure μ satisfying (2). Then, for λ_{ψ} ≥(d+1)/d

dim W_{K}(ψ) ≤ δ − α (1 − \frac{d+1}{λ_{ψ} d}) .

As a special case we obtain the following statement.

Corollary 2 Let K be a compact subset of R^{d} equipped with an absolutely α–friendly measure μ satisfying (2). Then, for τ ≥(d+1)/d

dim W_{K}(τ) ≤ δ − α (1 − \frac{d+1}{τ d}) .

Note that for τ > (d+1)/d we have that dim W_{K}(τ) < δ. Since μ is comparable to H^{δ} restricted to K, it follows that μ(W_{K}(τ)) = 0.

A general remark: For d ≥ 2, it is highly unlikely that either Theorem 1 or Theorem 2 are ever sharp. For instance, take the case that K := [0, 1]^{d} and μ is d–dimensional Lebesgue measure. It is easily verified that μ is absolutely α–friendly with α = 1. Thus, Theorem 1 implies that |W_{K}(ψ)|_{d} = 0 whenever

∑_{r=1}^{∞} r^{\frac{d+1}{d} −1} ψ(r) < ∞ .

So when d = 1 this coincides with the Lemma 1. However, for d ≥ 2 the above statement is weaker than that of the lemma. In view of Khintchines theorem one knows that the lemma is sharp; that is to say that if the sum in the lemma diverges then not only is |W_{K}(ψ)|_{d} > 0 but it is of full measure. It is probable that the theorems of this paper are sharp in the case d = 1.

1.2 The main example

The following statement which combines Theorems 2.2 and 8.1 of [2], shows that a large class of fractal measures are absolutely α–friendly and satisfy (2).

Theorem KLW Let {S_{1}, …, S_{k}} be an irreducible family of contracting self similarity maps of R^{d} satisfying the open set condition and let μ be the restriction of H^{δ} to its attractor K where δ := dim K. Then μ is absolutely α–friendly and satisfies (2).

Thus for the natural measures associated with self similar sets satisfying the open set condition, Theorems 1 and 2 are applicable. The simpliest examples of such sets include regular Cantor sets, the Sierpiński gasket and the von Kock curve. All the terminology except for ‘irreducible’ is pretty much standard – see for example [1, Chp.9]. The notion of irreducible introduced in [2, §2] avoids the natural obstruction that there is a finite collection of proper affine subspaces of R^{d} which is invariant under {S_{1}, …, S_{k}}.

2 Hausdorff measures and dimension

In this short section we define Hausdorff measure and dimension for completeness and in order to establish some notation. For ρ > 0, a countable collection {B_{i}} of Euclidean balls in R^{d} of radii r_{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} r_{i}^{s} : {B_{i}} is a ρ−cover of X} ,

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

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

and the Hausdorff dimension dim X of X by

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

Further details and alternative definitions of Hausdorff measure and dimension can be found in [1].

3 A covering lemma

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

Covering Lemma Let (Ω, d) be a metric space and B be a finite collection of balls with common radius r > 0. Then there exists a disjoint sub-collection {B_{i}} such that

⋃_{B∈B} B ⊂ ⋃_{i} 3B_{i} .

Proof : Let S denote the set of centres of the balls in B. Choose c_{1} ∈ S and for k ≥ 1,

c_{k+1} ∈ S \ ⋃_{i=1}^{k} B(c_{i}, 2r)

as long as S \ ⋃_{i=1}^{k} B(c_{i}, 2r) ≠ ∅. Since #S is finite, there exists k_{1} ≤ #S such that

S ⊂ ⋃_{i=1}^{k_{1}} B(c_{i}, 2r) .

By construction, any ball B(c, r) in the original collection B is contained in some ball B(c_{i}, 3r) and since d(c_{i}, c_{j}) > 2r the chosen balls B(c_{i}, r) are clearly disjoint.

♠

4 Proof of Theorem 1

The proof of Theorem 1 is separated into two parts; d = 1 and d ≥ 2. We first deal with the slightly more difficult case of when d ≥ 2.

4.1 The case when d ≥ 2

We shall prove the theorem in the case that d = 2. Thus, K is compact subset of R^{2}. There are no difficulties and no new ideas are required in extending the proof to higher dimensions.

Step 1: Preliminaries. We are assuming that ∑ r^{α \frac{3}{2} −1}ψ(r)^{α} converges and since ψ is monotonic, it follows that

∑_{n=1}^{∞} (2^{n \frac{3}{2}} ψ(2^{n}))^{α} < ∞ .  (3)

Next notice, that without loss of generality we can assume that

ψ(2^{n}) < c 2^{−n \frac{3}{2}}  (4)

for any c > 0 and n sufficiently large. This is easy to see. Suppose on the contrary that there exists a sequence \{n_{i}\} such that ψ(2^{n_{i}}) ≥ c 2^{−n_{i} \frac{3}{2}} . Then

∑_{n=1}^{∞} (2^{n \frac{3}{2}} ψ(2^{n}))^{α} ≥ ∑_{i=1}^{∞} (2^{n_{i} \frac{3}{2}} ψ(2^{n_{i}}))^{α} ≥ c^{α} ∑_{n=1}^{∞} 1 = ∞ ,

and this contradicts (3).

Step 2: The balls D_{n}. For n ∈ N, let D_{n} denote a generic ball with centre in K and of radius

r_{n} := \frac{1}{24} 2^{− \frac{3}{2} (n+1)} .

In view of the covering lemma and the fact that K is compact, there exists a finite, disjoint collection D_{n} of balls D_{n} with centers in K such that

⋃_{D_{n}} 3D_{n} ⊃ K .

Note that since μ is doubling, we have that

∑_{D_{n}} μ(3D_{n}) ≤ ∑_{D_{n}} μ(4D_{n}) ≤ D^{2} ∑_{D_{n}} μ(D_{n}) = D^{2} μ(⋃_{D_{n}}^{∘} D_{n}) ≤ D^{2} μ(K) . (5)

Next, consider a ball 3D_{n} where D_{n} ∈ D_{n}. Suppose there is a rational point p/q := (p_{1}/q, p_{2}/q) such that

x ∈ B(p/q, 2ψ(q)) and  2^{n} ≤ q < 2^{n+1} ,  (6)

for some point x = (x_{1}, x_{2}) in 3D_{n}. By (4) and using the fact that ψ is decreasing, it follows that for n sufficiently large p/q ∈ 6D_{n}. Now assume there are three or more such rational points satisfying (6); take any three (p_{1}/q, p_{2}/q), (p′_{1}/q′, p′_{2}/q′) and (p′′_{1}/q′′, p′′_{2}/q′′). In view of the denominator constraint, the rational points are necessarily distinct. Suppose for the moment that they do not lie on a line (i.e. a (d − 1)–dimensional hyperplane) and form the triangle Δ sub-tended by them. Twice the area (2–dimensional Lebesgue measure) 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, by (6)

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

Trivially,

area(6D_{n}) = \frac{π}{16} 2^{−3(n+1)} < \frac{1}{4} 2^{−3(n+1)} .

Thus area(Δ) > area(6D_{n}) and this is impossible since Δ ⊂ 6D_{n}. The upshot of this is that the triangle in question can not exist and so if there are two or more rational points satisfying (6) then they must lie on a line L.

Step 3: The finale. For n ∈ N, let

A_{n} := ⋃_{2^{n}≤q<2^{n+1}} ⋃_{p∈Z^{2}} B (p/q, 2ψ(q)) .

By definition, W_{K}(ψ) ⊂ lim sup_{n→∞} A_{n} ∩ K. It follows, that for n sufficiently large

μ(A_{n}) := μ(A_{n} ∩ K) = μ (A_{n} ∩ ⋃_{D_{n}} 3D_{n})
≤ ∑_{D_{n}} μ (3D_{n} ∩ L^{(ϵ)})  ϵ := 2 ψ(2^{n})
≪ (2^{n \frac{3}{2}} ψ(2^{n}))^{α} ∑_{D_{n}} μ(3D_{n})  μ is absolutely α–decaying
≪ (2^{n \frac{3}{2}} ψ(2^{n}))^{α} μ(K)  by (5).

Hence, by (3)

∑ μ(A_{n} ∩ K) = ∑ μ(A_{n}) ≪ ∑ (2^{n \frac{3}{2}} ψ(2^{n}))^{α} < ∞

and the Borel-Cantelli lemma implies that μ(lim sup_{n→∞} A_{n}) = 0. Thus, μ(W_{K}(ψ)) is zero as required.

♠

4.2 The case when d = 1

We prove the stronger statement which does not assume that μ is doubling – see the remark straight after the statement of Theorem 1. Although the proof below is basically the same as that in [3], we have decided to include a sketch for the sake of completeness and more importantly to bring out the true nature of the ‘triangle’ argument and the role of hyperplanes in the d ≥ 2 proof above. Clearly we can assume (3) and (4) with \frac{3}{2} replaced by 2. Consider rationals p/q with 2^{n} ≤ q < 2^{n+1}. For any two such rationals, notice that

|\frac{p}{q} − \frac{p′}{q′} | ≥ \frac{1}{qq′} > 2^{−2(n+1)} := 2 r_{n} .

Thus, any interval of length 2r_{n} can contain at most one rational. In particular, the intervals B(p/q, r_{n}) are disjoint. With A_{n} as in Step 3 above, we have that for n sufficiently large

μ(A_{n}) := μ(A_{n} ∩ K) = ∑_{2^{n}≤q<2^{n+1}} ∑_{p∈Z} μ(B(p/q, ϵ r_{n}))  ϵ r_{n} := 2 ψ(2^{n})

≪ (2^{2n}ψ(2^{n}))^{α} μ(K)  by (1) and disjointness.

As in Step 3 above, the Borel-Cantelli lemma implies the desired statement.

♠

5 Proof of Theorem 2

To a certain extent the proof of Theorem 2 is similar to that of Theorem 1. Again, we shall only establish the theorem in the case that d = 2.

Step 1: Preliminaries. Without loss of generality we can assume that ψ(r) → 0 as r → ∞. Suppose that this was not the case. Then W_{K}(ψ) = K by Dirichlet’s theorem and so H^{s}(W_{K}(ψ)) > 0 for any s ≤ δ - see Remark 2 straight after the statement of Theorem 2.

Without loss of generality we can assume that s > δ − α. If this where not the case then the sum in the statement of the theorem cannot possibly converge.

Since ψ is monotonic, the convergence of the sum in the statement of the theorem is equivalent to

∑_{n=1}^{∞} 2^{nα \frac{3}{2}} ψ(2^{n})^{α+s−δ} < ∞ .  (7)

Finally, notice that since s > δ−α, we can assume (4) without any loss of generality. Otherwise, (7) would be contradicted.

Step 2: A good ρ–cover for W_{K}(ψ). For n ∈ N, let D_{n} be the disjoint collection of balls D_{n} as defined in Step 2 of §4.1. Since the collection is disjoint and μ satisfies (2), we have that for n sufficiently large

#D_{n} × 2^{− \frac{3}{2} (n+1)δ} ≍ ∑_{D_{n}} μ(D_{n}) = μ(⋃_{D_{n}}^{∘} D_{n}) ≤ μ(K) .

Thus, for n sufficiently large

#D_{n} ≪ 2^{\frac{3}{2} (n+1)δ} .  (8)

Now put ϵ := 2ψ(2^{n}) and fix some ball D_{n} ∈ D_{n}. In view of the covering lemma, there exists a finite disjoint collection C(D_{n}) of balls B_{n}(ψ) with centers in K such that

⋃_{C(D_{n})} 3B_{n}(ψ) ⊃ 3 D_{n} ∩ L^{(ϵ)}

and

⋃_{C(D_{n})}^{∘} B_{n}(ψ) ⊂ 6 D_{n} ∩ L^{(2ϵ)}

The latter together with the fact that μ is absolutely α–decaying implies that

#C(D_{n}) × ψ(2^{n})^{δ} ≍ ∑_{C(D_{n})} μ(B_{n}(ψ)) = μ( ⋃_{C(D_{n})}^{∘} B_{n}(ψ))

≤ μ(6 D_{n} ∩ L^{(2ϵ)}) ≪ (2^{\frac{3}{2} (n+1)}ψ(2^{n}))^{α} 2^{− \frac{3}{2} (n+1)δ} .

Thus, for n sufficiently large

#C(D_{n}) ≪ (2^{\frac{3}{2} (n+1)}ψ(2^{n}))^{α−δ} .  (9)

Now with A_{n} defined as in Step 3 of §4.1, we have that

A_{n} ∩ K ⊂ ⋃_{D_{n}∈D_{n}} ⋃_{B_{n}(ψ)∈C(D_{n})}3 B_{n}(ψ) .

In particular, for each k ∈ N the collection

{3 B_{n}(ψ) : B_{n}(ψ) ∈ C(D_{n}), D_{n} ∈ D_{n} and n = k, k + 1, ⋯} ,

is a ρ–cover for W_{K}(ψ) with ρ = ρ(k) := 3 ψ(2^{k}).

Step 3: The finale. Let ρ = ρ(k) := 3 ψ(2^{k}). Step 2 together with the definition of s–dimensional Hausdorff measure implies that

H_{ρ}^{s}(W_{K}(ψ)) ≤ ∑_{n=k}^{∞} ∑_{D_{n}∈D_{n}} ∑_{B_{n}(ψ)∈C(D_{n})}(3 ψ(2^{n}))^{s} .

Thus, in view of (8) and (9), it follows that for k sufficiently large

H_{ρ}^{s}(W_{K}(ψ)) ≪ ∑_{n=k}^{∞} 2^{nα \frac{3}{2}} ψ(2^{n})^{α+s−δ} .

This together with (7) implies that

H_{ρ}^{s}(W_{K}(ψ)) → 0 as ρ → 0 (k → ∞) ,

and so H^{s}(W_{K}(ψ)) = 0 as required.

♠

Acknowledgements: SV would like to thank Barak Weiss for sending him the preprint [2] from which this work has materialized and for numerous e-mail conversations. He would also like to thank Ayesha and Iona for making 2003 such a fulfilling year – may there be many more. Much appreciated and thanks again girls. Finally he would like to thank the wonderful staff at ‘Bright Beginnings’ who have been absolutely fantastic with Ayesha and Iona – in particular their best friend Claire Kerner and Angela Foley for taking them back.

References

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

[2] D. Kleinbock, E. Lindenstrauss and B. Weiss : On fractal measures and Diophantine approximation. Selecta Mathematica. To appear.

[3] B. Weiss : Almost no points on a Cantor set are very well approximable. Proc. R. Soc. London. 457 (2001), 949–952.

[4] B. Weiss : Dynamics on parameter spaces: submanifold and fractal subset questions, in Rigidity in Dynamics and Geometry: M. Burger and A. Iozzi (eds.). Springer, (2002).

Andy Pollington: Department of Mathematics, Brigham Young University, Provo, UT 84602, USA.

e-mail: : andy@math.byu.edu

Sanju L. Velani: Department of Mathematics, University of York, Heslington, York, YO1 5DD, England.

e-mail: slv3@york.ac.uk