Fractional parts of powers of negative rationals

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