Poissonian pair correlation in higher dimensions

Abstract

Let (xn)n=1∞ be a sequence on the torus T (normalized to length 1). A sequence (xn) is said to have Poissonian pair correlation if, for all s>0,limN→∞⁡1N#{1≤m≠n≤N:|xm−xn|≤sN}=2s. It is known that this implies uniform distribution of the sequence (xn). Hinrichs, Kaltenböck, Larcher, Stockinger & Ullrich extended this result to higher dimensions and showed that sequences (xn) in [0,1]d that satisfy, for all s>0,limN→∞⁡1N#{1≤m≠n≤N:‖xm−xn‖∞≤sN1/d}=(2s)d are also uniformly distributed. We prove the same result for the extension by the Euclidean norm: if a sequence (xn) in Td satisfies, for all s>0,limN→∞⁡1N#{1≤m≠n≤N:‖xm−xn‖2≤sN1/d}=ωdsd, where ωd is the volume of the unit ball, then (xn) is uniformly distributed. Our approach shows that Poissonian Pair Correlation implies an exponential sum estimate that resembles and implies the Weyl criterion.