关于$n^2α$小数部分的三重相关
On the triple correlations of fractional parts of $n^2α$
摘要：
对于固定的$α\in [0,1]$，考虑扩张正方形$α, 4α, 9α, \dots, N^2α\, $模$1$的集合$S_{α,N}$。Rudnick和Sarnak指出，对于Lebesgue几乎所有这样的$α$，$S_{α,N}$的间隙分布与泊松模型一致（在$N$趋于无穷大的极限下）。在本文中，我们证明了一个新的估计的三重相关与这个问题，建立了一个渐近表达式的三阶矩的$S_{α,N}$的元素的数量在一个随机区间的长度$L/N$，假设$L > N^{1/4+\varepsilon}$。$1/4$的阈值基本上小于$1/2$的阈值（其是将由原始差异估计给出的阈值）。与对相关理论不同的是，对于递增整数的非缺项序列$(a_n)_{n=1}^{\infty} $，关于膨胀$(αa_n \, \text{mod } 1)_{n=1}^{\infty} $的三重相关知之甚少。这部分是由于三重相关函数的二阶矩难以控制，因此涉及方差界的标准技术不适用。我们绕过这一僵局，使用的鲁德尼克-Sarnak-Zaharescu和希斯-布朗，它连接的三重相关函数的一些模块化计数问题的作品的启发参数。在附录中，我们评论的差异和相关函数之间的关系，回答一个问题的Steinerberger。
英文摘要：
For fixed $α\in [0,1]$, consider the set $S_{α,N}$ of dilated squares $α, 4α, 9α, \dots, N^2α\, $ modulo $1$. Rudnick and Sarnak conjectured that for Lebesgue almost all such $α$ the gap-distribution 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 $L > N^{1/4+\varepsilon}$. 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 \, \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 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.
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