论文标题
关于算术随机波的淋巴结量的差异
On the variance of the nodal volume of arithmetic random waves
论文作者
论文摘要
Rudnick and Wigman（Ann。HenriPoincaré，2008; Arxiv：Math-PH/0702081）猜想，$ d $ d $ d $ d $ - d $ - d $ d的算术随机波的量的差异是$ o（e/\ nathcal {n} n}） $ \ Mathcal {n} $是对应于$ e $的特征空间的维度。当$ d = 2 $和$ d = 3 $时，先前的结果已经以更强大的渐近性确定了这一点。在此简短的说明中，我们证明了$ o（e/\ Mathcal {n}^{1+α（d）-ε}）$的上限，对于任何$ε> 0 $和$ d \ geq 4 $，其中$α（d）$是正的，并且倾向于零d $。当$ d \ geq 5 $ $ \ ell^{2} $ - Bourgain and Demeter的解耦猜想时，使用当前方法（最多$ε$）是最好的节能（最多$ε$）。
Rudnick and Wigman (Ann. Henri Poincaré, 2008; arXiv:math-ph/0702081) conjectured that the variance of the volume of the nodal set of arithmetic random waves on the $d$-dimensional torus is $O(E/\mathcal{N})$, as $E\to\infty$, where $E$ is the energy and $\mathcal{N}$ is the dimension of the eigenspace corresponding to $E$. Previous results have established this with stronger asymptotics when $d=2$ and $d=3$. In this brief note we prove an upper bound of the form $O(E/\mathcal{N}^{1+α(d)-ε})$, for any $ε>0$ and $d\geq 4$, where $α(d)$ is positive and tends to zero with $d$. The power saving is the best possible with the current method (up to $ε$) when $d\geq 5$ due to the proof of the $\ell^{2}$-decoupling conjecture by Bourgain and Demeter.
