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For fixed $\alpha \in [0,1]$, consider the set $S_{\alpha,N}$ of dilatedsquares $\alpha, 4\alpha, 9\alpha, \dots, N^2\alpha \, $ modulo $1$. Rudnickand Sarnak conjectured that for Lebesgue almost all such $\alpha$ thegap-distribution of $S_{\alpha,N}$ is consistent with the Poisson model (in thelimit as $N$ tends to infinity). In this paper we prove a new estimate for thetriple correlations associated to this problem, establishing an asymptoticexpression for the third moment of the number of elements of $S_{\alpha,N}$ ina random interval of length $L/N$, provided that $L > N^{1/4+\varepsilon}$. Thethreshold of $1/4$ is substantially smaller than the threshold of $1/2$ (whichis the threshold that would be given by a na\"{i}ve discrepancy estimate).  Unlike the theory of pair correlations, rather little is known about triplecorrelations of the dilations $(\alpha a_n \, \text{mod } 1)_{n=1}^{\infty} $for a non-lacunary sequence $(a_n)_{n=1}^{\infty} $ of increasing integers.This is partially due to the fact that second moment of the triple correlationfunction is difficult to control, and thus standard techniques involvingvariance bounds are not applicable. We circumvent this impasse by using anargument inspired by works of Rudnick--Sarnak--Zaharescu and Heath-Brown, whichconnects the triple correlation function to some modular counting problems.  In an appendix we comment on the relationship between discrepancy andcorrelation functions, answering a question of Steinerberger.

On the triple correlations of fractional parts of $n^2\alpha$