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Bernoulli decomposition and arithmetical independence between sequences
Published online by Cambridge University Press: 14 January 2020
Abstract
In this paper, we study the set $$\begin{eqnarray}A=\{p(n)+2^{n}d~\text{mod}~1:n\geq 1\}\subset [0,1],\end{eqnarray}$$
$p$ 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.
MSC classification
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- Original Article
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- Creative Commons
- This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
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- © The Author(s) 2020. Published by Cambridge University Press
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