Poissonian Pair Correlation and Discrepancy

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 $$ \frac{1}{N}\# \left\{ 1 \leq m \neq n \leq N: |x_m - x_n| \leq \frac{s}{N} \right\} \leq (1+δ)2s \qquad \mbox{for all} \hspace{6pt} 1 \leq s \leq (8/δ)\sqrt{\log{N}},$$ 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 $$N^2{D}_N^5\lesssim \frac{2}{N}{\int }_0^{N/2}{\left|\frac{1}{N}\#\left\{1\le m\ne n\le N:|x_m-x_n|\le \frac{s}{N}\right\}-2s\right|}^{2}ds\lesssim N^2{D}_N^2,$$ 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.