On the triple correlations of fractional parts of $n^2α$

Abstract

For fixed α ∈ [0, 1], consider the set S α,N of dilated squares α, 4α, 9α, . . . , N 2 α modulo 1. Rudnick and Sarnak conjectured that for Lebesgue almost all such α the gapdistribution of S α,N is consistent with the Poisson model (in the limit as N tends to infinity). In this paper we prove a new estimate for the triple correlations associated to this problem, establishing an asymptotic expression for the third moment of the number of elements of S α,N in a random interval of length L/N , provided that L > N 1/4+ε . The threshold of 1/4 is substantially smaller than the threshold of 1/2 (which is the threshold that would be given by a naïve discrepancy estimate).Unlike the theory of pair correlations, rather little is known about triple correlations of the dilations (αa n mod 1) ∞ n=1 for a non-lacunary sequence (a n ) ∞ n=1 of increasing integers. This is partially due to the fact that second moment of the triple correlation function is difficult to control, and thus standard techniques involving variance bounds are not applicable. We circumvent this impasse by using an argument inspired by works of Rudnick-Sarnak-Zaharescu and Heath-Brown, which connects the triple correlation function to some modular counting problems.In an appendix we comment on the relationship between discrepancy and correlation functions, answering a question of Steinerberger.3 We should remark that Rudnick-Sarnak phrased their result in terms of the pair correlation function, rather than explicitly mentioning W α,L,N , but the results are equivalent.

Poissonian correlations of higher orders

“…This is shown implicitly in [10,Thm 3(i)] and also follows from Proposition 9 later in the text. Regarding correlations of higher orders, some relevant results are shown in [27] for the triple correlations of sequences of the form (n 2 α) n∈N and for values of the length s > 0 that lie in a range that depends on N.…”

“…We wish to establish a relation between C * k ([0, a], s, N) defined in (27) and the correlation counting function C * k (s, N) relevant to the sequence (z n ) n∈N . Since we need to specify to which sequence the counting function refers to, from now on we write C * k ((z n ) n∈N , s, N) for the function C * k that refers to (z n ) n∈N , and when we do not state which sequence the correlation function C * k refers to, it will be understood that it refers to (x n ) n∈N .…”

A pair correlation problem, and counting lattice points with the zeta function

“…4 A similar argument appears at the end of [32], where it is used to show that the L 2 approach fails to work in the case of the triple correlation of (n 2 α)n; cf. also [39] 5 It might be difficult to spot at a quick glance, so we briefly comment on where the speed of growth of (xn)n was used in our argument in Sections 5 and 7. There is a term |P (0)| 2 N 1+ε/4 coming from the contribution of values of t near the origin to the integral.…”

“…In the setting of Theorem 2 we can control the number of small differences xm − xn efficiently because of the particularly simple structure of the sequence. The relevant equations are (32) and (39).…”

Long-Range Correlations of Sequences Modulo 1

“…Then if we can appropriately bound these Weyl sums we can show that the error is small. It should be noted that if the discrepancy of a sequence goes to 0 fast enough, then one can prove that the long-range correlations converge to the Poissonian limit for τ < 1 2 − ǫ for all ǫ > 0 (see [TW20] for details on this relation). The methodology in this paper, when applied to our examples, will improve on this 'naïve' bound.…”