Integral powers of numbers in small intervals modulo $1$: The cardinality gap phenomenon

Abstract

This paper deals with the distribution of $αζ^{n} \bmod 1$, where $α\neq 0,ζ>1$ are fixed real numbers and $n$ runs through the positive integers. Denote by $\Vert.\Vert$ the distance to the nearest integer. We investigate the case of $αζ^{n}$ all lying in prescribed small intervals modulo $1$ for all large $n$, with focus on the case $\Vertαζ^{n}\Vert \leq ε$ for small $ε>0$. We are particularly interested in what we call cardinality gap phenomena. For example for fixed $ζ>1$ and small $ε>0$ there are at most countably many values of $α$ such that $\Vertαζ^{n}\Vert \leq ε$ for all large $n$, whereas larger $ε$ induces an uncountable set. We investigate the value of $ε$ at which the gap occurs. We will pay particular attention to the case of algebraic and, more specific, rational $ζ>1$. Results concerning Pisot and Salem numbers such as some contribution to Mahler's $3/2$-problem are implicitly deduced. We study similar questions for fixed $α\neq 0$ as well.