Fractional parts of powers of negative rationals

TL;DR

This paper investigates the distribution of fractional parts of powers of negative rationals, establishing bounds on how tightly these fractional parts can cluster in the unit interval.

Contribution

It provides a new lower bound on the length of intervals containing the fractional parts of sequences generated by powers of negative rationals.

Findings

The sequence's fractional parts are not contained in any interval shorter than a specific bound.

The bound depends on the numerator and denominator of the rational base.

Results apply to irrational initial values or when the denominator exceeds one.

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 $(1+q/p-q^2/p^2)/p$.