There is no Khintchine threshold for metric pair correlations

TL;DR

This paper demonstrates that there is no simple Khintchine-like criterion based on additive energy to determine the metric pair correlation property of sequences, by constructing a counterexample sequence.

Contribution

It proves that no convergence/divergence criterion based solely on additive energy can fully characterize the metric pair correlation property.

Findings

Constructed a sequence with large additive energy that still has the metric pair correlation property.

Showed the non-existence of a Khintchine-type criterion for metric pair correlations.

Provided a counterexample to previous conjectures linking additive energy and pair correlation.

Abstract

We consider sequences of the form $\left(a_{n} \alpha\right)_{n}$ mod 1, where $\alpha\in\left[0,1\right]$ and where $\left(a_{n}\right)_{n}$ is a strictly increasing sequence of positive integers. If the asymptotic distribution of the pair correlations of this sequence follows the Poissonian model for almost all $\alpha$ in the sense of Lebesgue measure, we say that $(a_n)_n$ has the metric pair correlation property. Recent research has revealed a connection between the metric theory of pair correlations of such sequences, and the additive energy of truncations of $(a_n)_{n}$. Bloom, Chow, Gafni and Walker speculated that there might be a convergence/divergence criterion which fully characterises the metric pair correlation property in terms of the additive energy, similar to Khintchine's criterion in the metric theory of Diophantine approximation. In the present paper we give a negative answer to such speculations, by showing that such a criterion does not exist. To this end, we construct a sequence $(a_n)_n$ having large additive energy which, however, maintains the metric pair correlation property.