Some negative results related to Poissonian pair correlation problems

Abstract

We say that a sequence (xn)n∈N in [0,1) has Poissonian pair correlations if limN→∞1N#1≤l≠m≤N:‖xl−xm‖≤sN=2sfor every s≥0. The aim of this article is twofold. First, we will establish a gap theorem which allows to deduce that a sequence (xn)n∈N of real numbers in [0,1) having a certain weak gap structure, cannot have Poissonian pair correlations. This result covers a broad class of sequences, e.g., Kronecker sequences, the van der Corput sequence and more generally LS-sequences of points and digital (t,1)-sequences. Additionally, this theorem enables us to derive negative pair correlation results for sequences of the form ({anα})n∈N, where (an)n∈N is a strictly increasing sequence of integers with maximal order of additive energy, a notion that plays an important role in many fields, e.g., additive combinatorics, and is strongly connected to Poissonian pair correlation problems. These statements are not only metrical results, but hold for all possible choices of α.Second, in this note we study the pair correlation statistic for sequences of the form, xn={bnα},n=1,2,3,…, with an integer b≥2, where we choose α as the Stoneham number or as an infinite de Bruijn word. We will prove that both instances fail to have the Poissonian property.