Density of some sequences modulo $1$

Artūras Dubickas¹

¹ Department of Mathematics and Informatics Vilnius University Naugarduko 24 Vilnius LT-03225, Lithuania

Colloquium Mathematicum, Tome 128 (2012) no. 2, pp. 237-244

Résumé

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 $1 \leq n \leq N$.

Détail

DOI : 10.4064/cm128-2-9

Keywords: recently cilleruelo kumchev luca shparlinski proved each integer geq sequence fractional parts infty everywhere dense interval prove similar result pisot numbers salem numbers alpha each each sufficiently large every subinterval length contains least fractional part alpha where nonconstant polynomial mathbb integer satisfying leq leq