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arXiv · 1611.10111

The dimension of irregular set in parameter space

Lixuan Zheng·Min Wu·Bing Li
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Abstract

For any real number $β>1$. The $n$th cylinder of $β$ in the parameter space $\{β\in \mathbb{R}: β>1\}$ is a set of real numbers in $(1,\infty)$ having the same first $n$ digits in their $β$-expansion of $1$, denote by $I^P_n(β)$. We study the quantities which describe the growth of the length of $I^P_n(β)$. The Huasdorff dimension of the set of given growth rate of the length of $I^P_n(β)$ will be determined in this paper.

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Lixuan Zheng, Min Wu, Bing Li. 2016-11-30. The dimension of irregular set in parameter space. https://arxiv.org/abs/1611.10111

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