Poissonian Pair Correlation and Discrepancy

A sequence $(x_n{)}_{n=1}^{\mathrm{\infty }}$ 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. $$\underset{N\to \mathrm{\infty }}{\lim }\frac{1}{N}\mathrm{\#}\{1\le m\mathrm{\ne }n\le N:\mathrm{\mid }{x}_m-x_n\mathrm{\mid }\le \frac{s}{N}\}=2salmostsurely\mathrm{.}$$ We show that being close to Poissonian pair correlation for few values of $s$ is enough to deduce global regularity statements: if, for some~$0<\delta <1\mathrm{/}2$, a set of points $\{x_1,\dots ,x_N\}$ satisfies $$\frac{1}{N}\mathrm{\#}\{1\le m\mathrm{\ne }n\le N:\mathrm{\mid }{x}_m-x_n\mathrm{\mid }\le \frac{s}{N}\}\le (1+\delta )2sforall1\le s\le (8\mathrm{/}\delta )\sqrt{\log N},$$ 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 $$N^2{D}_N^5\lesssim \frac{2}{N}{\int }_0^{N\mathrm{/}2}{\mid \frac{1}{N}\mathrm{\#}\{1\le m\mathrm{\ne }n\le N:\mathrm{\mid }{x}_m-x_n\mathrm{\mid }\le \frac{s}{N}\}-2s\mid }^{2}ds\lesssim N^2{D}_N^2,$$ where the lower is bound is conditioned on $D_N\gtrsim N^{-1\mathrm{/}3}$. 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.