Additive energy, discrepancy and Poissonian $k$-level correlation

TL;DR

This paper constructs a probabilistic sequence with high additive energy and Poissonian $k$-level correlation for all $k \, \geq \, 2$, answering a longstanding open question in the field.

Contribution

It demonstrates the existence of sequences with both high additive energy and Poissonian $k$-level correlation for all $k \, \geq \, 2$, using a probabilistic approach.

Findings

Random perturbations of sequences with small discrepancy yield Poissonian $k$-level correlation almost surely.

The construction confirms the existence of well spaced sequences with high additive energy and Poissonian correlations.

The probabilistic method may be useful for future sequence constructions with similar properties.

Abstract

$k$-level correlation is a local statistic of sequences modulo 1, describing the local spacings of $k$-tuples of elements. For $k = 2$ this is also known as pair correlation. We show that there exists a well spaced increasing sequence of reals with additive energy of order $N^3$ and Poissonian $k$-level correlation for all integers $k \geq 2$, answering in the affirmative a question raised by Aistleitner, El-Baz, and Munsch. The construction is probabilistic, and so we do not obtain a specific sequence satisfying this condition. To prove this, we show that random perturbations of a sequence with small discrepancy gives, almost surely, a sequence with Poissonian $k$-level correlation, a fact which may be of independent interest.