Weak Poissonian box correlations of higher order

TL;DR

This paper introduces $(k,\beta)$-Poissonian box correlations as a unified framework linking uniform distribution, discrepancy theory, and gap statistics, enriching the theory of Poissonian correlations. It develops two equivalent formulations via $R_{k,\beta}$ and through the $G_\beta$/$H_\beta$ integral framework, proving a key equivalence (and partial equivalence for $k=2$) that connects box-based counts to functional correlation notions. The authors show that low-discrepancy sequences satisfy $(k,\beta)$-Poissonian box correlations for all $0<\beta<1$, while sequences with restricted gap structures obstruct $(k,1)$-box correlations, highlighting a nuanced trade-off between discrepancy, gaps, and higher-order correlations. They also outline open questions and potential generalizations, including monotonicity in $\beta$, equivalence of definitions, and higher-dimensional extensions, suggesting a fertile direction for future research.

Abstract

Poissonian pair correlations have sparked interest within the mathematical community, because of their number theoretic properties, and their connections to quantum physics and probability theory, particularly uniformly distributed random numbers. Rather recently, several generalizations of the concept have been introduced, including weak Poissonian pair correlations and $k$-th order Poissonian correlations. In this paper, we propose a new generalized concept called $(k,m,β)$-Poissonian box correlations. We study their properties and more specifically their relation to uniform distribution theory, discrepancy theory, random numbers and gap distributions.

Figures (1)

• Figure 1: The Kronecker sequence for $\alpha=\frac{1}{5}\sqrt{2}$. The gap structure can be observed on the left.

Theorems & Definitions (20)

• Theorem 1.1
• Theorem 1.2: hauke:poiss_corr_higher_order
• Theorem 1.3: hauke:poiss_corr_higher_order
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Proposition 2.1
• proof
• Proposition 2.2