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  • 天元讲堂(12.9):Multiple expansions of real numbers with digits set $\{0,1,q\}$
  • 浏览量:10 发布人: 发布时间:2018-12-06
  • 报告人: 李文侠教授 (华东师范大学)

    题目:Multiple expansions of real numbers  with  digits set $\{0,1,q\}$

    报告时间: 2018年12月9日(周日)下午 3:30-4:30

    报告地点: 数学二楼报告厅


    摘要:For $q>1$  we consider expansions in base $q$ over the alphabet $\{0,1,q\}$. Let ${\mathcal U}_q$ be the set of $x$ which have a unique $q$-expansions. For   $k=2, 3,\cdots,\aleph_0$ let ${\mathcal B}_k$ be the set of bases $q$ for which there exists $x$ having precisely $k$ different  $q$-expansions, and  for $q\in {\mathcal B}_k$ let ${\mathcal U}_q^{(k)}$ be the set of all such  $x$'s which have exactly $k$ different $q$-expansions. In this paper we show that

    \[

    {\mathcal B}_{\aleph_0}=[2,\infty)\quad\textrm{and}\quad

    {\mathcal B}_k=(q_c,\infty)\quad \textrm{for any}\quad k\ge 2,\]

     where

    $q_c\approx 2.32472$ is the appropriate root of

    $x^3-3x^2+2x-1=0$.

     Moreover,  we show that for any positive integer $k\ge 2$ and any  $q\in{\mathcal B}_{k}$ the Hausdorff dimensions of ${\mathcal U}_q^{(k)}$  and ${\mathcal U}_q$ are   the same, i.e.,

    $$

    \dim_H{\mathcal U}_q^{(k)}=\dim_H{\mathcal U}_q\quad\textrm{for any}\quad k\ge 2.

    $$

    Finally, we conclude that the set of $x$ having a continuum of $q$-expansions has full Hausdorff dimension.


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