On Exceptional Sets in the Metric Poissonian Pair Correlations problem

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.