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

TL;DR

This paper explores the distribution of powers of numbers modulo 1 within small intervals, revealing a cardinality gap phenomenon where the set of such numbers abruptly changes from countable to uncountable as the interval size varies.

Contribution

It introduces the concept of cardinality gap phenomena in the distribution of algebraic numbers' powers modulo 1 and analyzes the critical interval size for this transition.

Findings

For fixed $eta>1$, the set of $eta^n$ with small fractional parts is countable for small thresholds.

Larger thresholds lead to uncountably many such numbers, indicating a sharp transition.

Results have implications for Mahler's $3/2$-problem and special algebraic numbers like Pisot and Salem numbers.

Abstract

This paper deals with the distribution of $\alpha \zeta^{n} \bmod 1$, where $\alpha\neq 0,\zeta>1$ 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 $\Vert\alpha \zeta^{n}\Vert \leq \epsilon$ 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 $\Vert\alpha \zeta^{n}\Vert \leq \epsilon$ 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 $\alpha\neq 0$ as well.