Metric Poissonian Pair Correlation Property Part 2: The Case nlog n

@misc{tan2026metric,
  title = {{Metric Poissonian Pair Correlation Property Part 2: The Case \(n\log n\)}},
  author = {Bo Tan and Zhen Zhang and Qing-Long Zhou},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.22833},
  eprint = {2610.22833},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.22833},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Metric Poissonian Pair Correlation Property Part 2: The Case nlog n
AU  - Bo Tan
AU  - Zhen Zhang
AU  - Qing-Long Zhou
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.22833
UR  - https://arxiv.org/abs/2610.22833
AB  - We prove that (αnlog n)_n≥1 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.
ER  -