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arXiv · 2603.22045

Alternating geometric progressions modulo one and Sturmian words

Qing Lu·Weizhe Zheng
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Abstract

Let $b\ge 2$ be an integer. Using Sturmian words we describe all irrational real numbers $\xi$ such that the image in $\mathbb{R}/\mathbb{Z}$ of the sequence $(\xi (-b)^n)_{n\ge 0}$ is contained in an interval of length $b^{-1}+b^{-2}-b^{-3}$. In previous work (arXiv:2603.16794) we showed that the image cannot be contained in a shorter interval.

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Qing Lu, Weizhe Zheng. 2026-03-23. Alternating geometric progressions modulo one and Sturmian words. https://arxiv.org/abs/2603.22045

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