On exceptional sets in the metric Poissonian pair correlations problem

Let a n 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 A N = a n : n ≤ N , and a n 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 α a n 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.