Schmidt's game, fractals, and orbits of toral endomorphisms

TL;DR

This paper extends the understanding of the size and intersection properties of sets defined by non-converging orbits under matrix actions on tori, generalizing previous results to broader classes of matrices and sequences.

Contribution

It generalizes Dani's 1988 results to all points and matrices, and shows these sets intersect with fractals, also providing an alternative proof for a recent approximation theorem.

Findings

Sets have full Hausdorff dimension for all points and matrices.

Sets intersect with regular fractal subsets of Euclidean space.

Provides an alternative proof for a theorem on badly approximable systems.

Abstract

Given an integer nonsingular $n\times n$ matrix $M$ and a point $y \in \mathbb{R}^n/\mathbb{Z}^n$, consider the set $\tilde E(M,y)$ of vectors $x\in \mathbb{R}^n$ such that $y$ is not a limit point of the sequence $\{M^k x \mod \mathbb{Z}^n: k\in\mathbb{N}\}$. S.G. Dani showed in 1988 that whenever $M$ is semisimple and $y \in \mathbb{Q}^n/\mathbb{Z}^n$, the set $\tilde E(M,y)$ has full Hausdorff dimension. In this paper we strengthen this result, extending it to arbitrary $y \in \mathbb{R}^n/\mathbb{Z}^n$ and integer nonsingular $M$, and in fact replacing the sequence of powers of $M$ by any lacunary sequence of (not necessarily integer) $m\times n$ matrices. Furthermore, we show that sets of the form $\tilde E(M,y)$ and their generalizations always intersect with `sufficiently regular' fractal subsets of $\mathbb{R}^n$. As an application we give an alternative proof of a recent result of Einsiedler and Tseng on badly approximable systems of affine forms.