PreprintCC BY 4.0arXiv
Poissonian Pair Correlation for Bernoulli Orbits
Abstract
Let α, β∈ 𝕋 = ℝ/ℤ, and let (ξn)n ≥ 1 be independent random variables taking the values α and β with equal probability. We study the random walk Xn = ξ1 + ⋯ + ξn pmod 1 and present a deterministic Diophantine condition on (α, β) that implies almost sure Poisson pair correlation. To the best of our knowledge, this is the first result providing explicit examples in this framework where the increment measure is purely discrete. The classical Fourier contraction method fails in this setting; a different approach is required. Our proof uses the parabolic geometry of returns to 1 and an exact covariance formula for overlapping blocks.
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BibTeX
@misc{tan2026poissonian,
title = {{Poissonian Pair Correlation for Bernoulli Orbits}},
author = {Bo Tan and Zhen Zhang and Qing-Long Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15700},
eprint = {2610.15700},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15700},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Poissonian Pair Correlation for Bernoulli Orbits
AU - Bo Tan
AU - Zhen Zhang
AU - Qing-Long Zhou
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15700
UR - https://arxiv.org/abs/2610.15700
AB - Let α, β∈ 𝕋 = ℝ/ℤ, and let (ξ_n)_n ≥ 1 be independent random variables taking the values α and β with equal probability. We study the random walk X_n = ξ_1 + ⋯ + ξ_n pmod 1 and present a deterministic Diophantine condition on (α, β) that implies almost sure Poisson pair correlation. To the best of our knowledge, this is the first result providing explicit examples in this framework where the increment measure is purely discrete. The classical Fourier contraction method fails in this setting; a different approach is required. Our proof uses the parabolic geometry of returns to 1 and an exact covariance formula for overlapping blocks.
ER -