平坦环面的二次型与半经典本征函数假设
Quadratic forms and semiclassical eigenfunction hypothesis for flat tori
摘要：
设$Q(X)$是任意具有判别式$D$的整本原正定二次型，且在$k$变量中，其中$k\geq4$ .对于任意整数$n$，利用$n$，$k$和$D$给出了$Q(X)=n$的整数解个数的一个上界.作为推论，我们对Rudnick和Lester关于平坦环面$\mathbb{T}^d$上本征函数的正交基在单个本征空间上的小尺度等分布的猜想给出了明确的回答，对于$d\geq 5$ .我们的主要定理的另一个应用程序给出了一个尖锐的上限$A_{d}(n,t)$，一些代表性的正定二次形式$Q(x,y)=nx^2+2txy+ny^2$作为平方和的$d\geq 5$线性形式，其中$n- n^{\frac{1}{(d-1)}-o(1)}< t < n$。这个上界允许我们研究球面上整点的局部统计。
英文摘要：
Let $Q(X)$ be any integral primitive positive definite quadratic form with discriminant $D$ and in $k$ variables where $k\geq4$. We give an upper bound on the number of integral solutions of $Q(X)=n$ for any integer $n$ in terms of $n$, $k$ and $D$. As a corollary, we give a definite answer to a conjecture of Rudnick and Lester on the small scale equidistribution of orthonormal basis of eigenfunctions restricted to an individual eigenspace on the flat torus $\mathbb{T}^d$ for $d\geq 5$. Another application of our main theorem gives a sharp upper bound on $A_{d}(n,t)$, the number of representation of the positive definite quadratic form $Q(x,y)=nx^2+2txy+ny^2$ as a sum of squares of $d\geq 5$ linear forms where $n- n^{\frac{1}{(d-1)}-o(1)}< t < n$. This upper bound allows us to study the local statistics of integral points on sphere.
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