On a criterion of uniform distribution

TL;DR

This paper extends van der Corput's criterion for uniform distribution of sequences, providing new conditions involving weak monotonicity and second differences, with a proof based on Diophantine approximation.

Contribution

It introduces a new criterion for uniform distribution that relaxes previous conditions, using a different proof technique involving Diophantine approximation.

Findings

Sequence is uniformly distributed if weakly monotonic and satisfies second difference conditions.

Proof uses Diophantine approximation, differing from traditional exponential sum estimates.

Extends the theoretical understanding of criteria for uniform distribution.

Abstract

We give an extension of a criterion of van der Corput on uniform distribution of sequences. Namely, we prove that a sequence $x_n$ is uniformly distributed modulo 1 if it is weakly monotonic and satisfies the conditions $\Delta^2x_n\to 0,\quad n^2\Delta^2x_n\to \infty $. Our proof is straightforward and uses a Diophantine approximation by rational numbers, while van der Corput's approach is based on some estimates of exponential sums.