On exceptional sets in the metric Poissonian pair correlations problem

Let (an) n be a strictly increasing sequence of positive integers. Recentworks uncovered a close connection between the additive energy E (AN) 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 a. [ 0, 1] satisfying that (aan ) n with E (AN) = N3 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.