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Bernoulli decomposition and arithmetical independence between sequences
Ergodic Theory and Dynamical Systems ( IF 0.9 ) Pub Date : 2020-01-14 , DOI: 10.1017/etds.2019.117
HAN YU

In this paper, we study the set $$\begin{eqnarray}A=\{p(n)+2^{n}d~\text{mod}~1:n\geq 1\}\subset [0,1],\end{eqnarray}$$ where $p$ is a polynomial with at least one irrational coefficient on non-constant terms, $d$ is any real number and, for $a\in [0,\infty )$, $a~\text{mod}~1$ is the fractional part of $a$. With the help of a method recently introduced by Wu, we show that the closure of $A$ must have full Hausdorff dimension.

中文翻译:

伯努利分解和序列之间的算术独立性

在本文中,我们研究了集合$$\begin{eqnarray}A=\{p(n)+2^{n}d~\text{mod}~1:n\geq 1\}\subset [0,1],\end{eqnarray} $$在哪里$p$是在非常数项上具有至少一个无理系数的多项式,$d$是任何实数,并且,对于$a\in [0,\infty )$,$a~\text{mod}~1$是小数部分$a$. 借助 Wu 最近介绍的一种方法,我们证明了$澳元必须具有完整的 Hausdorff 维度。
更新日期:2020-01-14
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