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arXiv · 1508.00520

Dirichlet uniformly well-approximated numbers

Dong Han Kim·Lingmin Liao
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Abstract

Fix an irrational number $\theta$. For a real number $\tau >0$, consider the numbers $y$ satisfying that for all large number $Q$, there exists an integer $1\leq n\leq Q$, such that $\|n\theta-y\| 0$, the Haussdorff dimension of the set of these numbers is obtained and is shown to depend on the Diophantine property of $\theta$. It is also proved that with respect to $\tau$, the only possible discontinuous point of the Hausdorff dimension is $\tau=1$.

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Dong Han Kim, Lingmin Liao. 2015-08-03. Dirichlet uniformly well-approximated numbers. https://arxiv.org/abs/1508.00520

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