Poissonian correlation of higher order differences

Abstract

A sequence (xn)n=1∞ on the torus T exhibits Poissonian pair correlation if for all s>0,limN→∞⁡1N#{1≤m≠n≤N:|xm−xn|≤sN}=2s. It is known that this condition implies equidistribution of (xn). We generalize this result to four-fold differences: if for all s>0 we havelimN→∞⁡1N2#{1≤m,n,k,l≤N{m,n}≠{k,l}:|xm+xn−xk−xl|≤sN2}=2s then (xn)n=1∞ is equidistributed. This notion generalizes to higher orders, and for any k we show that a sequence exhibiting 2k-fold Poissonian correlation is equidistributed. In the course of this investigation we obtain a discrepancy bound for a sequence in terms of its closeness to 2k-fold Poissonian correlation. This result refines earlier bounds of Grepstad & Larcher and Steinerberger in the case of pair correlation, and resolves an open question of Steinerberger.