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HAUSDORFF DIMENSIONS OF SOME LIMINF SETS IN DIOPHANTINE APPROXIMATION

  • Bao-Wei Wang (a1), Zhi-Ying Wen (a2) and Jun Wu (a3)

Abstract

Let $Q$ be an infinite subset of $\mathbb{N}$ . For any ${\it\tau}>2$ , denote $W_{{\it\tau}}(Q)$ (respectively $W_{{\it\tau}}$ ) to be the set of ${\it\tau}$ well-approximable points by rationals with denominators in $Q$ (respectively in $\mathbb{N}$ ). We consider the Hausdorff dimension of the liminf set $W_{{\it\tau}}\setminus W_{{\it\tau}}(Q)$ after Adiceam. By using the tools of continued fractions, it is shown that if $Q$ is a so-called $\mathbb{N}\setminus Q$ -free set, the Hausdorff dimension of $W_{{\it\tau}}\setminus W_{{\it\tau}}(Q)$ is the same as that of $W_{{\it\tau}}$ , i.e.  $2/{\it\tau}$ .

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HAUSDORFF DIMENSIONS OF SOME LIMINF SETS IN DIOPHANTINE APPROXIMATION

  • Bao-Wei Wang (a1), Zhi-Ying Wen (a2) and Jun Wu (a3)

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