Quadratic forms and semiclassical eigenfunction hypothesis for flat tori

Abstract

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.