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

TL;DR

This paper establishes a link between Diophantine approximation and the lower Assouad dimension, showing that badly approximable points in certain self-affine fractals have Hausdorff dimension bounded below by the fractals' dynamical dimension, extending results beyond conformal cases.

Contribution

It introduces a novel connection between the lower Assouad dimension and badly approximable points in non-conformal fractals, advancing beyond previous conformal-only results.

Findings

Hausdorff dimension of badly approximable points bounded below by dynamical dimension

Full Hausdorff dimension of badly approximable points in self-affine sponges with equal dimensions

Extension of results to non-conformal fractals like Bedford-McMullen and Barański carpets

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. In particular, 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 Sierpi\'nski sponges/carpets (also known as Bedford-McMullen sponges/carpets) and the case of Bara\'nski 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.