Quantitative uniform distribution results for geometric progressions

TL;DR

This paper provides a precise quantitative analysis of the uniform distribution of fractional parts of geometric progressions, extending Koksma's classical theorem with exact asymptotic discrepancy estimates for typical values of the base.

Contribution

It offers an exact quantitative version of Koksma's theorem by calculating the asymptotic order of discrepancy for fractional parts of geometric progressions for almost all bases.

Findings

Derived explicit asymptotic discrepancy bounds

Extended uniform distribution results to arbitrary increasing sequences

Confirmed typical behavior for Lebesgue almost all bases

Abstract

By a classical theorem of Koksma the sequence of fractional parts $(\{x^n\})_{n \geq 1}$ is uniformly distributed for almost all values of $x$. In the present paper we obtain an exact quantitative version of Koksma's theorem, by calculating the precise asymptotic order of the discrepancy of $(\{\xi x^{s_n}\})_{n \geq 1}$ for typical values of $x>1$ (in the sense of Lebesgue measure). Here $\xi>0$ is an arbitrary constant, and $(s_n)_{n \geq 1}$ can be any increasing sequence of positive integers.