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The pair correlation function of multi-dimensional low-discrepancy sequences with small stochastic error terms

  • In any dimension d≥2, there is no known example of a low-discrepancy sequence which possesses Poisssonian pair correlations. This is in some sense rather surprising, because low-discrepancy sequences always have β-Poissonian pair correlations for all 0<β<1/d and are therefore arbitrarily close to having Poissonian pair correlations (which corresponds to the case β=1/d). In this paper, we further elaborate on the closeness of the two notions. We show that d-dimensional Kronecker sequences for badly approximable vectors α→ with an arbitrary small uniformly distributed stochastic error term generically have β=1/d-Poissonian pair correlations.

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Author:Anja Bettina Schmiedt, Christian Weiß
DOI:https://doi.org/10.1016/j.jnt.2023.12.011
Parent Title (English):Journal of Number Theory
Document Type:Article (peer reviewed)
Language:English
Publication Year:2024
Tag:Kronecker sequences; Poissonian pair correlations; Uniform distribution
Volume:259
First Page:422
Last Page:437