Dirichlet uniformly well-approximated numbers

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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Submitted on : Monday, August 21, 2017 - 10:50:39 AM
Last modification on : Thursday, January 11, 2018 - 6:12:17 AM

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  • HAL Id : hal-01182812, version 2
  • ARXIV : 1508.00520

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Dong Han Kim, Lingmin Liao. Dirichlet uniformly well-approximated numbers. 2017. ⟨hal-01182812v2⟩

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