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. Recent works uncovered a close connection between the additive energy Eleft( A_{N}right) of the cut-offs A_{N}=left{ a_{n},{:},,nle Nright} , and left( a_{n}right) _{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 alpha in left[ 0,1right] satisfying that left( leftlangle alpha a_{n}rightrangle right) _{n} with Eleft( A_{N}right) =Omega left( N^{3}right) 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.