Poissonian correlations of higher orders

TL;DR

This paper demonstrates that sequences with Poissonian correlations of any order are uniformly distributed and explores their connection to additive energy, providing a quantitative framework and extending known results to higher orders.

Contribution

It establishes that higher-order Poissonian correlations imply uniform distribution and extends the relationship between metric correlations and additive energy to these higher orders.

Findings

Sequences with Poissonian correlations are uniformly distributed.

Extended the connection between metric correlations and additive energy to higher orders.

Provided a quantitative description of Poissonian correlation phenomena.

Abstract

We show that any sequence $(x_n)_{n \in \mathbb{N}} \subseteq [0,1]$ that has Poissonian correlations of $k$-th order is uniformly distributed, also providing a quantitative description of this phenomenon. Additionally, we extend connections between metric correlations and additive energy, already known for pair correlations, to higher orders. Furthermore, we examine how the property of Poissonian $k$-th correlations is reflected in the asymptotic size of the moments of the function $F(t,s,N) = #\{n\leq N : \|x_n - t\| \leq s/(2N) \},\, t\in [0,1]. $