On pair correlation and discrepancy

Abstract

Abstract. We say that a sequence (xn) n≥1 in [0, 1) has Poissonian pair correlations iffor all s > 0. In this note we show that if the convergence in the above expression is-in a certain sense-fast, then this implies a small discrepancy for the sequence (xn) n≥1 . As an easy consequence it follows that every sequence with Poissonian pair correlations is uniformly distributed in [0, 1).

Poissonian pair correlation and discrepancy

“…This should be compared to a result of Grepstad & Larcher [5] that being δ−close to Poissonian pair correlation for s ∈ 1, . .…”

Pair correlation and equidistribution on manifolds

“…In two independent papers, Aistleitner, Lachmann and Pausinger [1] and Grepstad and Larcher [8] reversed the question and asked whether Poisson pair correlation (1.2) of a given sequence implies uniform distribution. The answer is yes, even under weaker hypotheses than (1.2), for sequences in the unit interval [1,8,18]. The same has been established for point sequences on higher-dimensional tori [10,19].…”

Pair correlation of sequences with maximal additive energy

“…Definition 1 Let · denote the distance to the nearest integer. A sequence (x n ) n∈N in [ 0, 1 ) is said to have (asymptotically) Poissonian pair correlations, if for each s > 0 the pair correlation function It is known that if a sequence (x n ) n∈N has Poissonian pair correlations, then it is uniformly distributed modulo 1, cf., [3,7,16]. The converse is not true in general.…”