Badly approximable points on self-affine sponges and the lower Assouad dimension

Tushar Das,Mariusz Urbański,Lior Fishman,David Simmons

doi:10.1017/etds.2017.42

Abstract

We highlight a connection between Diophantine approximation and the lower Assouad dimension by using information about the latter to show that the Hausdorff dimension of the set of badly approximable points that lie in certain non-conformal fractals, known as self-affine sponges, is bounded below by the dynamical dimension of these fractals. For self-affine sponges with equal Hausdorff and dynamical dimensions, the set of badly approximable points has full Hausdorff dimension in the sponge. Our results, which are the first to advance beyond the conformal setting, encompass both the case of Sierpiński sponges/carpets (also known as Bedford–McMullen sponges/carpets) and the case of Barański carpets. We use the fact that the lower Assouad dimension of a hyperplane diffuse set constitutes a lower bound for the Hausdorff dimension of the set of badly approximable points in that set.

Highlights

A point x ∈ Rd is said to be badly approximable if this inequality cannot be improved by more than a constant, i.e. if there exists a constant c > 0 such that, for any rational
Since BAd has full dimension, one expects its intersection with any fractal set J ⊆ Rd to have the same dimension as J, and this can be proven for certain broad classes of fractal sets J, see e.g. [4, 6, 12, 20]
The class of self-affine sponges is the generalization to higher dimensions of the class of self-affine carpets, which consists of subsets of R2 defined according to a certain recursive construction where each rectangle in the construction is replaced by the union of several rectangles contained in that rectangle

Summary

Introduction

A point x ∈ Rd is said to be badly approximable if this inequality cannot be improved by more than a constant, i.e. if there exists a constant c > 0 such that, for any rational. The reason that we call (2) reducible is that its limit set does not have any hyperplane diffuse subsets, meaning that Schmidt’s game cannot be used to deduce lower bounds on the dimension of its intersection with BAd ; see §2 and Proposition 3.4 for details.

Results

Conclusion