A note on sequences not having metric Poissonian pair correlations

TL;DR

This paper constructs sequences that lack metric Poissonian pair correlations despite having additive energies near the believed threshold, offering a simpler proof compared to previous work.

Contribution

It provides a new, simplified construction of sequences without MPPC with additive energies close to the critical threshold, extending prior results with a different approach.

Findings

Sequences without MPPC can have additive energies near the threshold

A simpler proof method is introduced for such sequences

The construction modifies Bourgain's approach

Abstract

The purpose of this note is to present a construction of sequences which do not have metric Poissonian pair correlations (MPPC) and whose additive energies grow at rates that come arbitrarily close to a threshold below which it is believed that all sequences have MPPC. A similar result appears in work of Lachmann and Technau and is proved using a totally different strategy. The main novelty here is the simplicity of the proof, which we arrive at by modifying a construction of Bourgain.