Poissonian Pair Correlation and Discrepancy

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.