Dimension Bound for Badly Approximable Grids

Abstract

Abstract We show that there exists a subset of full Lebesgue measure $V\subset \mathbb{R}^{n}$ such that for every ϵ > 0 there exists δ > 0 such that for any v ∈ V the dimension of the set of vectors w satisfying $$ \liminf_{k\to\infty} k^{1/n}\langle kv-w\rangle\geqslant \epsilon$$ (where 〈⋅〉 denotes the distance from the nearest integer) is bounded above by n − δ. 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.