Fractional parts of powers of negative rationals
Volume 224 / 2026
Acta Arithmetica 224 (2026), 387-394
MSC: Primary 11J54; Secondary 11J71, 68R15
DOI: 10.4064/aa260330-17-6
Published online: 10 September 2026
Abstract
We prove that for any real number $\xi \neq 0$ and any coprime integers $p \gt q\ge 1$, if $\xi $ is irrational or $q \gt 1$, then 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 $(1+q/p-q^2/p^2)/p$.
