Badly approximable points on non-linear carpets

TL;DR

This paper establishes that certain non-linear, non-conformal attractors in Diophantine approximation have a full intersection with badly approximable points, expanding understanding of their geometric and measure-theoretic properties.

Contribution

It identifies the first class of non-linear non-conformal attractors with full intersection property with badly approximable points and provides a formula for their Hausdorff dimension.

Findings

Non-linear non-conformal attractors intersect badly approximable points in full dimension.

A new formula for the Hausdorff dimension of these attractors is derived.

Answers a 2019 question by Das-Fishman-Simmons-Urbański.

Abstract

The badly approximable points in $\mathbb{R}^d$ are those for which Dirichlet's approximation theorem cannot be improved by more than a constant, that is, they are the points most difficult to approximate by rational vectors. An important problem in Diophantine approximation is to determine when the set of badly approximable points intersects a given set in full dimension. We find the first class of non-linear non-conformal attractors for which this full intersection property holds, thus answering a question of Das-Fishman-Simmons-Urba\'nski from 2019. We also provide a formula for the Hausdorff dimension of these attractors which is of independent interest.