On fractional parts of powers of real numbers close to 1

TL;DR

This paper investigates the fractional parts of powers of real numbers close to 1, establishing bounds on their distance from integers and analyzing the density of fractional parts for certain real numbers, with implications for number theory.

Contribution

It proves the existence of small epsilon where powers of (1+epsilon) stay away from integers by a specific bound and shows the set of such real numbers with non-dense fractional parts has full Hausdorff dimension.

Findings

Existence of epsilon with powers staying away from integers by a quantifiable bound

The set of real numbers with non-dense fractional powers has full Hausdorff dimension

Bounds are sharp up to a specific logarithmic factor

Abstract

We prove that there exist arbitrarily small positive real numbers $\epsilon$ such that every integral power $(1 + \vepsilon)^n$ is at a distance greater than $2^{-17} \epsilon |\log \vepsilon|^{-1}$ to the set of rational integers. This is sharp up to the factor $2^{-17} |\log \epsilon|^{-1}$. We also establish that the set of real numbers $\alpha > 1$ such that the sequence of fractional parts $(\{\alpha^n\})_{n \ge 1}$ is not dense modulo 1 has full Hausdorff dimension.