Poissonian pair correlation and discrepancy

A sequence (xn)n=1∞ on the torus T≅[0,1] is said to exhibit Poissonian pair correlation if the local gaps behave like the spacings of a Poisson random variable, i.e. limN→∞1N#1≤m≠n≤N:|xm−xn|≤sN=2salmost 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<δ<1∕2, a set of points x1,…,xN satisfies 1N#1≤m≠n≤N:|xm−xn|≤sN≤(1+δ)2sfor all1≤s≤(8∕δ)logN,then the discrepancy DN of the set satisfies DN≲δ1∕3+N−1∕3δ−1∕2. We also show that distribution properties are reflected in the global deviation from the Poissonian pair correlation N2DN5≲2N∫0N∕21N#1≤m≠n≤N:|xm−xn|≤sN−2s2ds≲N2DN2,where the lower bound is conditioned on DN≳N−1∕3. The proofs use a connection between exponential sums, the heat kernel on T and spatial localization. Exponential sum estimates are obtained as a byproduct. We also describe a connection to diaphony and several open problems.