Some connections between discrepancy, finite gap properties, and pair correlations

Abstract

A generic uniformly distributed sequence $$(x_n)_{n \in \mathbb {N}}$$ ( x n ) n ∈ 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 $$\alpha $$ α -pair correlations with $$0< \alpha < 1$$ 0 < α < 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 $$\alpha $$ α -pair correlations for $$0< \alpha <1$$ 0 < α < 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 $$\alpha $$ α -pair correlations for $$0< \alpha < 1$$ 0 < α < 1 .

Sequences with almost Poissonian pair correlations

“…so that the case α = 1 corresponds to the usual pair correlation statistic, see also [Wei22] for an even more general notion. A sequence is said to have weak correlations if lim N →∞ F α N (s) = 2s for all s ≥ 0.…”

“…This connection of gaps and the pair correlation statistic partially explains the current popularity of the topic and has recently been very explicitly addressed in several publications, see e.g. [LS20], [Wei22] and [Wei23]. But also independently of the more specific pair correlation statistic, the gap structure of sequences in [0, 1] d is extensively explored in the literature, see e.g.…”

“…It has been generalized in many different regards, including e.g. to higher dimensions, see [HKL + 19], to weak pair correlations, see [NP07,HZ21] and below, to a combination of both, see [Wei22], to compact Riemannian manifolds, see [Mar19], and to higher order correlations, see [HZ23].…”

“…Due to a result which has been proved independently in [ALP18] and [GL17] for Poissonian pair correlations (i.e. α = 1) and which has been later generalized to weak pair correlations in [Ste18,Coh21,Wei22], we know that a sequence in [0, 1] with α-weak Poissonian pair correlations is uniformly distributed. This central property transfers to the p-adic integers as well.…”

“…Also here, most ingredients from the proof in the real case, see [GL17] and [Wei22], can be turned into precise arguments in the p-adic setting. Finally, we discuss the role of Kronecker sequences, which are to the author's knownledge the only known examples of sequences possessing the optimal order of convergence in terms of the p-adic discrepancy.…”