A generalization of van der Corput's Difference Theorem

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 $h \in \mathbb{N}$.