论文标题
关于$ n^2α$的分数部分的三重相关性
On the triple correlations of fractional parts of $n^2α$
论文作者
论文摘要
对于[0,1] $中的固定$α\，请考虑扩大正方形的$ s_ {α，n} $ $α，4α，9α，\ dots，n^2α\，$ modulo $ 1 $。鲁德尼克（Rudnick）和萨尔纳克（Sarnak）猜想，对于勒布斯格（Lebesgue）而言，几乎所有这样的$α$ $ s_ {α，n} $的间隙分布与泊松模型一致（以$ n $为$ n $ to to Infinity）。在本文中，我们证明了与此问题相关的三重相关性的新估计值，以$ s_ {α，n} $的第三次渐近表达方式，前提是$ l/n $的$ s_ {α，n} $，但前提是$ l/n $，前提是$ l> n^{1/4+\ var epepsilon} $。 $ 1/4 $的门槛大大小于$ 1/2 $的门槛（这是由幼稚的差异估算给出的阈值）。 与配对相关的理论不同，关于非验证序列$（A_n）_ {n = 1}^{n = 1}^{\ infty Integers的扩张$（αA_n\，\ text {mod} 1）_ {n = 1}^{n = 1}^{\ infty iNDEGER。这部分是由于难以控制三重相关函数的第二刻，因此不适用涉及方差界限的标准技术。我们通过使用由Rudnick-Sarnak-Zaharescu和Heath-Brown的作品启发的参数来避免这种僵局，该参数将三重相关函数与某些模块化计数问题联系起来。 在附录中，我们评论差异和相关功能之间的关系，回答了Steinerberger的问题。
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.
