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.
Author: | Anja Bettina Schmiedt, Christian Weiß |
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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 |