On the triple correlations of fractional parts of

Abstract

For fixed $\alpha \in [0,1]$ , consider the set $S_{\alpha ,N}$ of dilated squares $\alpha , 4\alpha , 9\alpha , \dots , N^2\alpha \, $ modulo $1$ . Rudnick and Sarnak conjectured that, for Lebesgue, almost all such $\alpha $ the gap-distribution of $S_{\alpha ,N}$ 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 with this problem, establishing an asymptotic expression for the third moment of the number of elements of $S_{\alpha ,N}$ in a random interval of length $L/N$ , provided that $L> N^{1/4+\varepsilon }$ . The threshold of $\tfrac {1}{4}$ is substantially smaller than the threshold of $\tfrac {1}{2}$ (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 $(\alpha a_n \, \text {mod } 1)_{n=1}^{\infty } $ for a nonlacunary sequence $(a_n)_{n=1}^{\infty } $ of increasing integers. This is partially due to the fact that the 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, and Zaharescu, and Heath-Brown, which connects the triple correlation function to some modular counting problems. In Appendix B, we comment on the relationship between discrepancy and correlation functions, answering a question of Steinerberger.

Intermediate-scale statistics for real-valued lacunary sequences

“…We would like to stress the difference between (1•1) and some weaker notions of longrange Poissonian correlations, as studied, e.g., in [5,7,17]. Note that the number variance 2 N (L, α) can be expressed in terms of the pair correlation function.…”

“…An example of such a statistic is the number variance (the variance of the number of elements in random intervals, see the definition below in our setting), famously studied for the zeros of the Riemann zeta function, for which at small scales the number variance is consistent with that of the eigenvalues of random matrices drawn from the Gaussian unitary ensemble (GUE), whereas "saturation" occurs at larger scales (see [1]). In the context of sequences modulo one, only a few results have been established so far in the mesoscopic regime, mainly concerning the leading order asymptotics of the long-range correlations of the sequence x n = αn 2 (see [4,5,7,12,17]); nevertheless, important intermediate-scale statistics such as the number variance have largely remained unexplored. The aim of this paper is to study such statistics for real-valued lacunary sequences.…”

On the number of gaps of sequences with Poissonian pair correlations

“…Whether this sequence has Poissonian triple correlation for almost all α is still unknown, and seems to be a difficult problem (cf. [20]). Rudnick, Sarnak and Zaharescu [22] conjectured that for almost all α the sequence has Poissonian correlations of all orders, as well as exponentially distributed gaps (see also [19,Conjecture 1.1]), but this is widely out of reach with current methods.…”