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Abstract

Recently, Cilleruelo, Kumchev, Luca, Rué and Shparlinski proved that for each integer $a \geq 2$ the sequence of fractional parts $\{a^n/n\}_{n=1}^{\infty}$ is everywhere dense in the interval $[0,1]$. We prove a similar result for all Pisot numbers and Salem numbers $\alpha$ and show that for each $c>0$ and each sufficiently large $N$, every subinterval of $[0,1]$ of length $cN^{-0.475}$ contains at least one fractional part $\{Q(\alpha^n)/n\}$, where $Q$ is a nonconstant polynomial in $\mathbb Z[z]$ and $n$ is an integer satisfying $1 \leq n \leq N$.