The set of badly approximable vectors is strongly $C^1$ incompressible

TL;DR

This paper demonstrates that the set of badly approximable vectors in Euclidean space remains large in a strong sense under smooth transformations, by establishing its hyperplane absolute winning property and full Hausdorff dimension.

Contribution

The paper introduces a new variant of Schmidt's game and proves the badly approximable vectors are hyperplane absolute winning, extending previous results to more general sets and fractals with simpler proofs.

Findings

The set of badly approximable vectors has full Hausdorff dimension after certain transformations.

It is hyperplane absolute winning, implying robustness under smooth maps.

Results extend to intersections with fractals, maintaining positive dimension.

Abstract

We prove that the countable intersection of $C^1$-diffeomorphic images of certain Diophantine sets has full Hausdorff dimension. For example, we show this for the set of badly approximable vectors in $\mathbb{R}^d$, improving earlier results of Schmidt and Dani. To prove this, inspired by ideas of McMullen, we define a new variant of Schmidt's $(\alpha, \beta)$-game and show that our sets are hyperplane absolute winning (HAW), which in particular implies winning in the original game. The HAW property passes automatically to games played on certain fractals, thus our sets intersect a large class of fractals in a set of positive dimension. This extends earlier results of Fishman to a more general set-up, with simpler proofs.