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arXiv · 1609.06531

On the expansions of real numbers in two multiplicative dependent bases

Yann Bugeaud·Dong Han Kim
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Abstract

Let $r \ge 2$ and $s \ge 2$ be multiplicatively dependent integers. We establish a lower bound for the sum of the block complexities of the $r$-ary expansion and of the $s$-ary expansion of an irrational real number, viewed as infinite words on $\{0, 1, \ldots , r-1\}$ and $\{0, 1, \ldots , s-1\}$, and we show that this bound is best possible.

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Yann Bugeaud, Dong Han Kim. 2016-09-21. On the expansions of real numbers in two multiplicative dependent bases. https://arxiv.org/abs/1609.06531

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