2020/09/17 by Christoph Aistleitner, Aistleitner, Christoph, Daniel El-Baz +3 · 1 citation
Mathematics · #11J25 #11J71 (Secondary) #11J83 #11K06 #11M06 (Primary) 11B05 #Advanced Mathematical Identities #Analytic Number Theory Research #Benford’s Law and Fraud Detection #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2009.08184
openalex publication_date 2020/09/17 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
The pair correlation is a localized statistic for sequences in the unit\ninterval. Pseudo-random behavior with respect to this statistic is called\nPoissonian behavior. The metric theory of pair correlations of sequences of the\nform (an \α)n \≥ 1 has been pioneered by Rudnick, Sarnak and\nZaharescu. Here \α is a real parameter, and (an)n \≥ 1 is an\ninteger sequence, often of arithmetic origin. Recently, a general framework was\ndeveloped which gives criteria for Poissonian pair correlation of such\nsequences for almost every real number \α, in terms of the additive\nenergy of the integer sequence (an)n \≥ 1. In the present paper we\ndevelop a similar framework for the case when (an)n \≥ 1 is a sequence\nof reals rather than integers, thereby pursuing a line of research which was\nrecently initiated by Rudnick and Technau. As an application of our method, we\nprove that for every real number \θ>1, the sequence (n^\θ \α)n\n\≥ 1 has Poissonian pair correlation for almost all \α \∈\n\ℝ.\n