On Exceptional Sets in the Metric Poissonian Pair Correlations problem

TL;DR

This paper investigates the threshold of additive energy in sequences of integers that determines whether their fractional parts under multiplication by almost every real number exhibit Poissonian pair correlations, revealing new boundary cases.

Contribution

It establishes new bounds on the additive energy that influence the metric Poissonian pair correlation property for sequences.

Findings

Sequences with high additive energy lack PPC for almost every alpha.

Constructs sequences with specific energy growth where PPC fails on full measure sets.

Shows full Hausdorff dimension of non-PPC set for certain energy thresholds.

Abstract

Let $\left(a_{n}\right)_{n}$ be a strictly increasing sequence of positive integers, denote by $A_{N}=\left\{ a_{n}:\,n\leq N\right\} $ its truncations, and let $\alpha\in\left[0,1\right]$. We prove that if the additive energy $E\left(A_{N}\right)$ of $A_{N}$ is in $\Omega\left(N^{3}\right)$, then the sequence $\left(\left\langle \alpha a_{n}\right\rangle \right)_{n}$ of fractional parts of $\alpha a_{n}$ does not have Poissonian pair correlations (PPC) for almost every $\alpha$ in the sense of Lebesgue measure. Conversely, it is known that $E\left(A_{N}\right)=\mathcal{O}\left(N^{3-\varepsilon}\right)$, for some fixed $\varepsilon>0$, implies that $\left(\left\langle \alpha a_{n}\right\rangle \right)_{n}$ has PPC for almost every $\alpha$. This note makes a contribution to investigating the energy threshold for $E\left(A_{N}\right)$ to imply this metric distribution property. We establish, in particular, that there exist sequences $\left(a_{n}\right)_{n}$ with \[ E\left(A_{N}\right)=\Theta\left(\frac{N^{3}}{\log\left(N\right)\log\left(\log N\right)}\right) \] such that the set of $\alpha$ for which $\left(\alpha a_{n}\right)_{n}$ does not have PPC is of full Lebesgue measure. Moreover, we show that for any fixed $\varepsilon>0$ there are sequences $\left(a_{n}\right)_{n}$ with $E\left(A_{N}\right)=\Theta\left(\frac{N^{3}}{\log\left(N\right)\left(\log\log N\right)^{1+\varepsilon}}\right)$ satisfying that the set of $\alpha$ for which the sequence $\left(\bigl\langle\alpha a_{n}\bigr\rangle\right)_{n}$ does not have PPC is of full Hausdorff dimension.