On exceptional sets in the metric Poissonian pair correlations problem

Let (an)n be a strictly increasing sequence of positive integers. Recent works uncovered a close connection between the additive energy E(A _N ) of the cut-offs AN={an:n≤N}, and (an)n possessing metric Poissonian pair correlations which is a metric version of a uniform distribution property of “second order”. Firstly, the present article makes progress on a conjecture of Aichinger, Aistleitner, and Larcher; by sharpening a theorem of Bourgain which states that the set of α∈ [0 , 1] satisfying that (〈αan〉)n with E(A _N ) = Ω (N ³ ) does not have Poissonian pair correlations has positive Lebesgue measure. Secondly, we construct sequences with high additive energy which do not have metric Poissonian pair correlations, in a strong sense, and provide Hausdorff dimension estimates.