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

TL;DR

This paper investigates the pair correlation properties of multi-dimensional low-discrepancy sequences, demonstrating that certain Kronecker sequences with small stochastic errors exhibit Poissonian pair correlations close to the theoretical maximum.

Contribution

It shows that d-dimensional Kronecker sequences with badly approximable vectors and small stochastic errors have $eta=1/d$-Poissonian pair correlations, advancing understanding of their statistical properties.

Findings

Kronecker sequences with badly approximable vectors exhibit $eta=1/d$-Poissonian pair correlations.

Low-discrepancy sequences are close to having Poissonian pair correlations for all $eta<1/d$.

Sequences with small stochastic errors generically display these correlation properties.

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.