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arXiv · 1009.4528

On fractional parts of powers of real numbers close to 1

Yann Bugeaud·Nikolay Moshchevitin
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Abstract

We prove that there exist arbitrarily small positive real numbers $ε$ such that every integral power $(1 + \vepsilon)^n$ is at a distance greater than $2^{-17} ε|\log \vepsilon|^{-1}$ to the set of rational integers. This is sharp up to the factor $2^{-17} |\log ε|^{-1}$. We also establish that the set of real numbers $α> 1$ such that the sequence of fractional parts $(\{α^n\})_{n \ge 1}$ is not dense modulo 1 has full Hausdorff dimension.

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Yann Bugeaud, Nikolay Moshchevitin. 2010-09-23. On fractional parts of powers of real numbers close to 1. https://arxiv.org/abs/1009.4528

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