Metric Poissonian Pair Correlation Property Part 2: The Case \(n\log n\)

Abstract

We prove that $(\alpha n\log n)_{n\ge1}$ has Poissonian pair correlation for almost every real $\alpha$, solving a Problem posed by Aistleitner, El-Baz and Munsch [Open Problem 3, Geom. Funct. Anal. 2021]. Our main contribution is a new four-point counting estimate with a fixed power saving over $N^{7/3}$ at tolerance $N^{-1}$, valid on the ranges left after a first-moment and Borel--Cantelli argument. Its proof reduces the counting condition to the required proximity of an implicitly defined real root to an integer; a Fej\'er kernel majorizes this condition, and multidimensional van der Corput $A$- and $B$-processes supply the saving. A localized Mellin argument then yields the second-moment estimate needed for the metric conclusion.