A generalization of van der Corput's Difference Theorem

TL;DR

This paper generalizes van der Corput's Difference Theorem within uniform distribution theory by linking it to unitary operators with Lebesgue spectrum, leading to new results on sequence distribution and spectral characterizations.

Contribution

It introduces a broad generalization of van der Corput's Difference Theorem connecting uniform distribution with spectral properties of unitary operators, and provides new characterizations of sequence distributions.

Findings

Sequences with uniformly distributed differences imply distribution along Thue-Morse subsequences.

A variant of the theorem relates to unitary operators with continuous spectrum.

New criteria for joint uniform distribution in multidimensional sequences.

Abstract

We prove a generalization of van der Corput's Difference Theorem in the theory of uniform distribution by establishing a connection with unitary operators that have Lebesgue spectrum. This allows us to show, for example, that if $(x_n)_{n = 1}^\infty \subseteq [0,1]$ is such that $(x_{n+h}-x_n)_{n = 1}^\infty$ is uniformly distributed for all $h \in \mathbb{N}$, then $(x_{n_k})_{k = 1}^\infty$ is uniformly distributed, where $(n_k)_{k = 1}^\infty$ is an enumeration of the $1s$ in the classical Thue-Morse sequence. We also establish a variant of van der Corput's Difference Theorem that is connected to unitary operators with continuous spectrum. Lastly, we obtain a new characterization of those sequence $(x_n)_{n = 1}^\infty \subseteq [0,1]$ for which $(x_{n+h},x_n)_{n = 1}^\infty$ is uniformly distributed in $[0,1]^2$ for all $h \in \mathbb{N}$.