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

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