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Dimension bound for badly approximable grids. (arXiv:1706.09600v1 [math.DS])
来源于:arXiv
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 $\liminf_{k\to\infty} k^{1/n}<kv-w> \ge \epsilon$ (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. 查看全文>>