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 $k\ge 4$. 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 $d\ge 5$. 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 $Q(x,y)=n{x}^2+2txy+n{y}^2$ as a sum of squares of $d\ge 5$ 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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