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 $A_N=\left\{a_n:n\le N\right\}$ 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 ${\left(⟨\alpha a_n⟩\right)}_n$ 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 $E\left(A_N\right)=\mathcal{O}\left(N^{3-\epsilon }\right)$, for some fixed $\epsilon >0$, implies that ${\left(⟨\alpha a_n⟩\right)}_n$ 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 \[ E\left(A_{N}\right)=Θ\left(\frac{N^{3}}{\log\left(N\right)\log\left(\log N\right)}\right) \] 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 $\epsilon >0$ there are sequences ${\left(a_n\right)}_n$ with $E\left(A_N\right)=\Theta \left(\frac{N^3}{\log \left(N\right){\left(\log \log N\right)}^{1+\epsilon }}\right)$ satisfying that the set of $\alpha$ for which the sequence ${\left(⟨\alpha a_n⟩\right)}_n$ does not have PPC is of full Hausdorff dimension.