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

Abstract

This paper deals with the distribution of $\alpha {\zeta }^n\mathrm{mod}1$, where $\alpha \ne 0,\zeta >1$ 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 $\alpha {\zeta }^n$ all lying in prescribed small intervals modulo $1$ for all large $n$, with focus on the case $\Vert \alpha {\zeta }^n\Vert \le \epsilon$ for small $\epsilon >0$. We are particularly interested in what we call cardinality gap phenomena. For example for fixed $\zeta >1$ and small $\epsilon >0$ there are at most countably many values of $\alpha$ such that $\Vert \alpha {\zeta }^n\Vert \le \epsilon$ for all large $n$, whereas larger $\epsilon$ induces an uncountable set. We investigate the value of $\epsilon$ at which the gap occurs. We will pay particular attention to the case of algebraic and, more specific, rational $\zeta >1$. 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 $\alpha \ne 0$ as well.

17 pages. A. Dubickas thankfully pointed out to me that the initial version of Theorem 2.1 can be improved by a result of Dobrowolski