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arXiv · 1512.06935

On the expansions of real numbers in two integer bases

Yann Bugeaud·Dong Han Kim
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Abstract

Let $r$ and $s$ be multiplicatively independent positive integers. We establish that the $r$-ary expansion and the $s$-ary expansion of an irrational real number, viewed as infinite words on $\{0, 1, \ldots , r-1\}$ and $\{0, 1, \ldots , s-1\}$, respectively, cannot have simultaneously a low block complexity. In particular, they cannot be both Sturmian words.

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Yann Bugeaud, Dong Han Kim. 2016-12-12. On the expansions of real numbers in two integer bases. https://arxiv.org/abs/1512.06935

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