Fractional parts of powers of real algebraic numbers

TL;DR

This paper provides an effective lower bound on how close powers of a real algebraic number greater than one can get to integers, advancing understanding of their fractional parts.

Contribution

It introduces a new effective bound for the fractional parts of powers of real algebraic numbers, improving previous results in Diophantine approximation.

Findings

Established an explicit lower bound for the distance between powers of algebraic numbers and integers.

Enhanced the understanding of the distribution of fractional parts of algebraic powers.

Provided tools potentially useful for further research in number theory and Diophantine approximation.

Abstract

Let $\alpha$ be a real number greater than $1$. We establish an effective lower bound for the distance between an integral power of $\alpha$ and its nearest integer.