On the smallest gap in a sequence with Poisson pair correlations

TL;DR

This paper proves that sequences with Poisson pair correlations and average gap 1 must have at least one gap exceeding 1.5, refining previous bounds and advancing understanding of sequence spacing properties.

Contribution

It establishes a new lower bound on the largest gap in sequences with Poisson pair correlations, improving prior results.

Findings

Sequences with Poisson pair correlations have gaps of at least 1.5.

The proven lower bound is slightly above 1.5, specifically 3/2+10^{-9}.

The result refines previous bounds on sequence gaps.

Abstract

We prove that any increasing sequence of real numbers with average gap $1$ and Poisson pair correlations has some gap that is at least $3/2+10^{-9}$. This improves upon a result of Aistleitner, Blomer, and Radziwill.