On the fractional parts of certain sequences of $\xi \alpha^{n}$

TL;DR

This paper investigates the distribution of fractional parts of sequences formed by algebraic numbers raised to powers, establishing bounds and applications to Fourier decay of self-similar measures under Diophantine conditions.

Contribution

It provides new lower bounds on the occurrence of fractional parts in sequences involving algebraic numbers and extends results on Fourier decay rates of self-similar measures.

Findings

Established lower bounds for fractional parts within sequences.

Extended results on Fourier decay of self-similar measures.

Generalized previous theorems under Diophantine conditions.

Abstract

Assume that $\alpha>1$ is an algebraic number and $\xi\neq0$ is a real number. We are concerned with the distribution of the fractional parts of the sequence $(\xi \alpha^{n})$. Under various Diophantine conditions on $\xi$ and $\alpha$, we obtain lower bounds on the number $n$ with $1\leq n\leq N $ for which the fractional part of the sequence $(\xi \alpha^{n})_{n\geq1}$ fall into a prescribed region $I\subset [0,1]$, extending several results in the literature. As an application, we show that the Fourier decay rate of some self-similar measures is logarithmic, generalizing a result of Varj\'{u} and Yu.