Poissonian correlation of higher order differences

[摘要] A sequence (x(n))(n=1)(infinity) on the torus T exhibits Poissonian pair correlation if for all s >0, lim(N ->infinity) 1/N#{1 <= m not equal m not equal n <= M : vertical bar x(m) - x(n) vertical bar <= s/N} = 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 have lim(N ->infinity)1/N-2{1 <= m,n,k,l <= N} : {m,n}={k,l} : |x(m) + x(n) - x(k) - x(l)| < s/2s N-2 } = 2s then (xn)infinity 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. (C) 2020 Elsevier Inc. All rights reserved.

[发布日期] 2021-12-01

[关键词] Pair correlation;Difference sequences