Poissonian Pair Correlation and Discrepancy

TL;DR

This paper investigates how local pair correlation properties of sequences on the torus relate to their global distribution regularity, establishing bounds and connections with discrepancy and exponential sums.

Contribution

It demonstrates that partial Poissonian pair correlation implies global discrepancy bounds and links distribution properties to pair correlation deviations.

Findings

Local pair correlation controls global discrepancy.

Discrepancy bounds derived from partial Poissonian behavior.

Connection established between pair correlation, discrepancy, and exponential sums.

Abstract

A sequence $(x_n)_{n=1}^{\infty}$ on the torus $\mathbb{T} \cong [0,1]$ is said to exhibit Poissonian pair correlation if the local gaps behave like the gaps of a Poisson random variable, i.e. $$ \lim_{N \rightarrow \infty}{ \frac{1}{N} \# \left\{ 1 \leq m \neq n \leq N: |x_m - x_n| \leq \frac{s}{N} \right\}} = 2s \qquad \mbox{almost surely.}$$ We show that being close to Poissonian pair correlation for few values of $s$ is enough to deduce global regularity statements: if, for some~$0 < \delta < 1/2$, a set of points $\left\{x_1, \dots, x_N \right\}$ satisfies $$ \frac{1}{N}\# \left\{ 1 \leq m \neq n \leq N: |x_m - x_n| \leq \frac{s}{N} \right\} \leq (1+\delta)2s \qquad \mbox{for all} \hspace{6pt} 1 \leq s \leq (8/\delta)\sqrt{\log{N}},$$ then the discrepancy $D_N$ of the set satisfies $D_N \lesssim \delta^{1/3} + N^{-1/3}\delta^{-1/2}$. We also show that distribution properties are reflected in the global deviation from the Poissonian pair correlation $$ N^2 D_N^5 \lesssim \frac{2}{N}\int_{0}^{N/2} \left| \frac{1}{N}\# \left\{ 1 \leq m \neq n \leq N: |x_m - x_n| \leq \frac{s}{N} \right\} - 2s \right|^2 ds \lesssim N^2 D_N^2,$$ where the lower is bound is conditioned on $D_N \gtrsim N^{-1/3}$. The proofs use a connection between exponential sums, the heat kernel on $\mathbb{T}$ and spatial localization. Exponential sum estimates are obtained as a byproduct. We also describe a connection to diaphony and several open problems.