On Exceptional Sets in the Metric Poissonian Pair Correlations problem

Let ${\left(a_n\right)}_n$ be a strictly increasing sequence of positive integers, denote by $A_N=\{a_n:n\le N\}$ its truncations, and let $\alpha \in [0,1]$. We prove that if the additive energy $E\left(A_N\right)$ of $A_N$ is in $\mathrm{\Omega }\left(N^3\right)$, 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 $\left(\left\langle \alpha a_{n}\right\rangle \right){n}$hasPPCforalmostevery$\alpha$. 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 $$E\left(A_N\right)=\mathrm{\Theta }\left(\frac{N^3}{\log \left(N\right)\log (\log N)}\right)$$ 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 $E\left(A{N}\right)=\Theta\left(\frac{N^{3}}{\log\left(N\right)\left(\log\log N\right)^{1+\varepsilon}}\right)$satisfyingthatthesetof$\alpha$ for which the sequence ${(⟨\alpha a_n⟩)}_n$ does not have PPC is of full Hausdorff dimension.