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arXiv · 2610.22833

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

Bo Tan·Zhen Zhang·Qing-Long Zhou
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Abstract

We prove that $(αn\log n)_{n\ge1}$ has Poissonian pair correlation for almost every real $α$, 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 Fejér 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.

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Bo Tan, Zhen Zhang, Qing-Long Zhou. 2026-10-16. Metric Poissonian Pair Correlation Property Part 2: The Case \(n\log n\). https://arxiv.org/abs/2610.22833

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