Pair correlation of sequences $(\lbrace a_n \alpha \rbrace)_{n \in \mathbb{N}}$ with maximal order of additive energy

TL;DR

The paper proves that sequences with maximal additive energy do not exhibit Poissonian pair correlations when multiplied by any real number, sharpening previous results in the field.

Contribution

It establishes a sharp criterion linking additive energy of sequences to their pair correlation properties, extending Bourgain's earlier work.

Findings

Sequences with maximal additive energy lack Poissonian pair correlations for all real multipliers.

The result sharpens the understanding of the relationship between additive structure and pair correlations.

Provides a definitive negative answer for sequences with maximal additive energy regarding Poissonian behavior.

Abstract

We show for sequences $\left(a_{n}\right)_{n \in \mathbb{N}}$ of distinct positive integers with maximal order of additive energy, that the sequence $\left(\left\{a_{n} \alpha\right\}\right)_{n \in \mathbb{N}}$ does not have Poissonian pair correlations for any $\alpha$. This result essentially sharpens a result obtained by J. Bourgain on this topic.