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arXiv · 2211.09891

The Pair Correlation Function of Multi-Dimensional Low-Discrepancy Sequences with Small Stochastic Error Terms

Anja Schmiedt·Christian Weiß
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Abstract

In any dimension $d \geq 2$, 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 $0 < β< \tfrac{1}{d}$ and are therefore arbitrarily close to having Poissonian pair correlations (which corresponds to the case $β= \tfrac{1}{d}$). In this paper, we further elaborate on the closeness of the two notions. We show that $d$-dimensional Kronecker sequences for badly approximable vectors $\vecα$ with an arbitrary small uniformly distributed stochastic error term generically have $β= \tfrac{1}{d}$-Poissonian pair correlations.

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Anja Schmiedt, Christian Weiß. 2023-03-23. The Pair Correlation Function of Multi-Dimensional Low-Discrepancy Sequences with Small Stochastic Error Terms. https://arxiv.org/abs/2211.09891

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