2022/11/17 by Anja Schmiedt, Schmiedt, Anja, Christian Weiß +1
Mathematics · #FOS: Mathematics #Mathematical Approximation and Integration #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2211.09891
openalex publication_date 2022/11/17 · openalex created_date 2022/11/28 · openalex updated_date 2026/07/28
In any dimension d ≥ 2, there is no known example of a low-discrepancy sequence which possess Poisssonian pair correlations. This is in some sense rather surprising, because low-discrepancy sequences always have β-Poissonian pair correlations for all 0 < β< \tfrac1d and are therefore arbitrarily close to having Poissonian pair correlations (which corresponds to the case β= \tfrac1d). 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= \tfrac1d-Poissonian pair correlations.