A note on sequences not having metric Poissonian pair correlations

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.

Automated summary

The note presents a construction of sequences that do not have metric Poissonian pair correlations (MPPC) and have additive energies that grow close to a threshold; similar result appears in the work of Lachmann and Technau but is proved using a different strategy. The main novelty lies in the simplicity of the proof, achieved by modifying a construction of Bourgain.