Hausdorff dimension and uniform exponents in dimension two

Abstract

In this paper we prove the Hausdorff dimension of the set of (nondegenerate) singular two-dimensional vectors with uniform exponent µ ∈ (1/2, 1) is 2(1 − µ) when µ ≥ √ 2/2, whereas for µ < √ 2/2 it is greater than 2(1 − µ) and at most (3 − 2µ)(1 − µ)/(1 + µ + µ 2 ). We also establish that this dimension tends to 4/3 (which is the dimension of the set of singular two-dimensional vectors) when µ tends to 1/2. These results improve upon previous estimates of R. Baker, joint work of the first author with M. Laurent, and unpublished work of M. Laurent. We also prove a lower bound on the packing dimension that is strictly greater than the Hausdorff dimension for µ ≥ 0.565 . . . . Definition 1. Let m, n be positive integers and A a real n × m matrix. The matrix A is badly approximable if there exists a positive constant c such that the system of inequalitieshas no solution x in Z m for any X ≥ 1.Definition 2. Let m, n be positive integers and A a real n × m matrix. We say that Dirichlet's Theorem can be improved for the matrix A if there exists a positive constant c < 1 such that the system of inequalities (1.1) has a solution x in Z m for any sufficiently large X.If the subgroup G = AZ m + Z n of R n generated by the m rows of the matrix t A (here and below, t M denotes the transpose of a matrix M) together with Z n has rank strictly less than m + n, then there exists x in Z m with |x| ∞ arbitrarily large, such that Ax = 0 and, consequently, for any real number w and any sufficiently large X > 1, the system of inequalitieshas a solution x in Z m . In several of the questions considered below, we have to exclude this degenerate situation, thus we are led to introduce the set M * n,m (R) of n × m matrices for which the associated subgroup G has rank m + n.When m = n = 1, that is, when A = (ξ) for some irrational real number ξ, it is not difficult to show that Dirichlet's Theorem can be improved if, and only if, ξ is badly approximable (or, equivalently, ξ has bounded partial quotients in its continued fraction expansion); see [19] and [13] for a precise statement. Furthermore, by using the theory of continued fractions, one can prove that, for any irrational real number ξ, there are arbitrarily large integers X such that the system of inequalities xξ|| ≤ 1 2X and 0 < x ≤ X (1.2) has no integer solutions; see Proposition 2.2.4 of [5].Since the set of badly approximable numbers has Lebesgue measure zero and Hausdorff dimension 1, this implies that the set of 1 × 1 matrices A for which Dirichlet's Theorem can be improved has Lebesgue measure zero and Hausdorff dimension 1. The latter assertion has been extended as follows.Theorem A. For any positive integers m, n, the set of real n×m matrices for which Dirichlet's Theorem can be improved has mn-dimensional Lebesgue measure zero and Hausdorff dimension mn.The first assertion of Theorem A has been established by Davenport and Schmidt [14] when min{m, n} = 1. According to Kleinbock and Weiss [21], their proof can be generalized to n × m matrices. Actually, a more general re...

A variational principle in the parametric geometry of numbers, with applications to metric Diophantine approximation

“…Theorem 1.9 implies that dim H (Sing 1,2 (ω)) < dim P (Sing 1,2 (ω)) for all ω ∈ (1/2, 1). This answers the first part of [1,Problem 7] in the affirmative.…”

“…In this paper, we announce a proof that their conjecture is correct. We will also show that the packing dimension of Sing(m, n) is the same as its Hausdorff dimension, thus answering a question of Bugeaud, Cheung, and Chevallier [1,Problem 7]. To summarize: Theorem 1.1.…”

“…The notion of singularity (in the sense of Diophantine approximation) was introduced by Khintchine, first in 1937 in the setting of simultaneous approximation [11], and later in 1948 in the more general setting of matrix approximation [12]. 1 Since then this notion has been studied within Diophantine approximation and allied fields, see Moshchevitin's 2010 survey [13]. An m × n matrix A is called singular if for all ε > 0, there exists Q ε such that for all Q ≥ Q ε , there exist integer vectors p ∈ Z m and q ∈ Z n such that Aq + p ≤ εQ −n/m and 0 < q ≤ Q.…”

“…It would be desirable to give precise formulas for the Hausdorff and packing dimensions of Sing m,n (ω) in terms of ω, m, and n, see e.g. [1,Problem 2]. However, this appears quite challenging at the present juncture, though we have made significant progress towards this question which we will describe in the next section.…”

Singular Vectors on Fractals and Projections of Self-similar Measures

“…Question 1.1 (Problem 6, [BCC18]). What is the dimension of the set of vectors in Sing(2) whose coordinates belong to Cantor's middle thirds set?…”