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arXiv · 1706.09600

Dimension bound for badly approximable grids

Seonhee Lim·Nicolas de Saxcé·Uri Shapira
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Abstract

We show that for almost any vector $v$ in $\mathbb{R}^n$, for any $ε>0$ there exists $δ>0$ such that the dimension of the set of vectors $w$ satisfying $\liminf_{k\to\infty} k^{1/n} \ge ε$ (where $<\cdot>$ 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.

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Seonhee Lim, Nicolas de Saxcé, Uri Shapira. 2017-06-29. Dimension bound for badly approximable grids. https://arxiv.org/abs/1706.09600

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