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- W2553039180 abstract "Given an integer $Nge 2$ and a real number ${beta}>1$, let $Gamma_{{beta},N}$ be the set of all $x=sum_{i=1}^infty {d_i}/{{beta}^i}$ with $d_iin{0,1,cdots,N-1}$ for all $ige 1$. The infinite sequence $(d_i)$ is called a ${beta}$-expansion of $x$. Let $mathbf{U}_{{beta},N}$ be the set of all $x$'s in $Gamma_{{beta},N}$ which have unique ${beta}$-expansions. We give explicit formula of the Hausdorff dimension of $mathbf{U}_{{beta},N}$ for ${beta}$ in any admissible interval $[{beta}_L,{beta}_U]$, where ${{beta}_L}$ is a purely Parry number while ${{beta}_U}$ is a transcendental number whose quasi-greedy expansion of $1$ is related to the classical Thue-Morse sequence. This allows us to calculate the Hausdorff dimension of $U{N}$ for almost every $beta>1$. In particular, this improves the main results of G{a}bor Kall{o}s (1999, 2001). Moreover, we find that the dimension function $f({beta})=dim_Hmathbf{U}_{{beta},N}$ fluctuates frequently for ${beta}in(1,N)$." @default.
- W2553039180 created "2016-11-30" @default.
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- W2553039180 date "2014-01-24" @default.
- W2553039180 modified "2023-10-12" @default.
- W2553039180 title "On the Hausdorff dimension of unique beta expansions" @default.
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