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发表时间:2018-12-12 阅读次数:50次
报告题目: Multiple expansions of real numbers with digits set $\set{0,1,q}$
报 告 人:李文侠教授
报告人所在单位:华东师范大学
报告日期:2018-12-12 星期三
报告时间:16:00-17:00
报告地点:光华楼东主楼1801
  
报告摘要:
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. This is a joint work with K.Dajani, K. Jiang and D.R. Kong.

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本年度学院报告总序号:274

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