Fractional parts of powers of negative rationals

Abstract

We prove that for any real number $\xi \ne 0$ and any coprime integers $p>q\ge 1$ 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 $(1+q/p-q^2/p^2)/p$.

9 pages. v4: fixed typos