Dimension bound for badly approximable grids

Abstract

We show that for almost any vector $v$ in ${\mathbb{R}}^n$, for any $\epsilon >0$ there exists $\delta >0$ such that the dimension of the set of vectors $w$ satisfying ${\mathrm{liminf}}_{k\to \infty }{k}^{1/n}<kv-w>\ge \epsilon$ (where $<\cdot >$ denotes the distance from the nearest integer), is bounded above by $n-\delta$. This result is obtained as a corollary of a discussion in homogeneous dynamics and the main tool in the proof is a relative version of the principle of uniqueness of measures with maximal entropy.

25 pages