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arXiv · 2006.15842

Badly approximable numbers, Kronecker's theorem, and diversity of Sturmian characteristic sequences

Dmitry Badziahin·Jeffrey Shallit
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Abstract

We give an optimal version of the classical ``three-gap theorem'' on the fractional parts of $n θ$, in the case where $θ$ is an irrational number that is badly approximable. As a consequence, we deduce a version of Kronecker's inhomogeneous approximation theorem in one dimension for badly approximable numbers. We apply these results to obtain an improved measure of sequence diversity for characteristic Sturmian sequences, where the slope is badly approximable.

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Dmitry Badziahin, Jeffrey Shallit. 2020-06-29. Badly approximable numbers, Kronecker's theorem, and diversity of Sturmian characteristic sequences. https://arxiv.org/abs/2006.15842

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