Quadratic forms and semiclassical eigenfunction hypothesis for flat tori

TL;DR

This paper establishes bounds on solutions to quadratic forms and applies these results to eigenfunction distribution on flat tori and representation problems, advancing understanding of eigenfunction equidistribution and integral point statistics.

Contribution

It provides new upper bounds on solutions to quadratic forms and confirms a conjecture on eigenfunction equidistribution on high-dimensional flat tori.

Findings

Upper bounds on solutions to quadratic forms in multiple variables.

Resolution of Rudnick and Lester's conjecture on eigenfunction distribution.

Sharp bounds on representations of quadratic forms as sums of squares.

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.