Fractional parts of powers of negative rationals

Qing Lu; Weizhe Zheng

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arXiv · 2603.16794

Fractional parts of powers of negative rationals

Qing Lu·Weizhe Zheng

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Abstract

We prove that for any real number $\xi\neq 0$ and any coprime integers $p>q\ge1$ such that $\xi$ is irrational or $q>1$, the image in $\mathbb{R}/\mathbb{Z}$ of the sequence $(\xi (-p/q)^n)_{n\ge 0}$ is not contained in any interval of length less than $(1+q/p-q^2/p^2)/p$.

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Qing Lu, Weizhe Zheng. 2026-03-17. Fractional parts of powers of negative rationals. https://arxiv.org/abs/2603.16794

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