PreprintCC BY 4.0arXiv
Poissonian Pair Correlation and Discrepancy Rates
Abstract
We prove the existence of a sequence in [0,1) with Poissonian pair correlation (PPC) and optimal-order star discrepancy DN*=O((log N)/N), and further give a deterministic construction by applying a conditional-expectation rule to a nested dyadic random model. We also show that PPC imposes no universal discrepancy decay rate: for every positive sequence ρN→0, there exists a PPC sequence with DN*/ρN→∞. Under suitable regularity and growth assumptions, we realize prescribed discrepancy asymptotics, including DN*∼ cβN−β for every 0<β<1, with cβ>0. We further construct a PPC sequence satisfying DN*asymp(log N)/N for all sufficiently large N.
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BibTeX
@misc{tan2026poissonian,
title = {{Poissonian Pair Correlation and Discrepancy Rates}},
author = {Bo Tan and Zhen Zhang and Qing-Long Zhou},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06041},
eprint = {2611.06041},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06041},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Poissonian Pair Correlation and Discrepancy Rates
AU - Bo Tan
AU - Zhen Zhang
AU - Qing-Long Zhou
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06041
UR - https://arxiv.org/abs/2611.06041
AB - We prove the existence of a sequence in [0,1) with Poissonian pair correlation (PPC) and optimal-order star discrepancy D_N^*=O((log N)/N), and further give a deterministic construction by applying a conditional-expectation rule to a nested dyadic random model. We also show that PPC imposes no universal discrepancy decay rate: for every positive sequence ρ_N→0, there exists a PPC sequence with D_N^*/ρ_N→∞. Under suitable regularity and growth assumptions, we realize prescribed discrepancy asymptotics, including D_N^*∼ c_βN^−β for every 0<β<1, with c_β>0. We further construct a PPC sequence satisfying D_N^*asymp(log N)/N for all sufficiently large N.
ER -