Metrical results on the distribution of fractional parts of powers of real numbers

TL;DR

This paper investigates the distribution properties of fractional parts of powers of real numbers, providing new metrical results and generalizing to sequences of functions with specific growth and regularity conditions.

Contribution

It introduces new metrical results on fractional parts distribution and extends the analysis to general sequences of functions beyond powers.

Findings

New metrical distribution results for fractional parts of powers

Generalization to sequences of functions with growth and regularity conditions

Enhanced understanding of distribution properties in number theory

Abstract

Denote by {$\times$} the fractional part. We establish several new metrical results on the distribution properties of the sequence ({x n }) n$\ge$1. Many of them are presented in a more general framework, in which the sequence of functions (x $\rightarrow$ x n) n$\ge$1 is replaced by a sequence (fn) n$\ge$1 , under some growth and regularity conditions on the functions fn.