Some Connections Between Discrepancy, Finite Gap Properties, and Pair Correlations

TL;DR

This paper explores the relationship between discrepancy, finite gap properties, and pair correlations in sequences, extending known results and establishing new links, especially for low-discrepancy sequences and their correlation properties.

Contribution

It generalizes bounds on discrepancy for sequences with alpha-pair correlations, and proves that low-discrepancy sequences have alpha-pair correlations, linking finite gap properties to pair correlation behavior.

Findings

Generalized discrepancy bounds for alpha-pair correlations.

Proved low-discrepancy sequences have alpha-pair correlations.

Re-established finite gap property implications for van der Corput and Kronecker sequences.

Abstract

A generic uniformly distributed sequence $(x_n)_{n \in \mathbb{N}}$ in $[0,1)$ possesses Poissonian pair correlations (PPC). Vice versa, it has been proven that a sequence with PPC is uniformly distributed. Grepstad and Larcher gave an explicit upper bound for the discrepancy of a sequence given that it has PPC. As a first result, we generalize here their result to the case of $\alpha$-pair correlations with $0 < \alpha < 1$. Since the highest possible level of uniformity is achieved by low-discrepancy sequences it is tempting to assume that there are examples of such sequences which also have PPC. Although there are no such known examples, we prove that every low-discrepancy sequence has at least $\alpha$-pair correlations for $0 < \alpha <1$. According to Larcher and Stockinger, the reason why many known classes of low-discrepancy sequences fail to have PPC is their finite gap property. In this article, we furthermore show that the discrepancy of a sequence with the finite gap property plus a condition on the distribution of the different gap lengths can be estimated. As a concrete application of this estimation, we re-prove the fact that van der Corput and Kronecker sequences are low-discrepancy sequences. Consequently, it follows from the finite gap property that these sequences have $\alpha$-pair correlations for $0 < \alpha < 1$.