# Diophantine equations and ergodic theorems

In the paper by Akos Magyar, Diophantine Equations and Ergodic Theorems, one states in page 923 the following theorem:

Theorem 1: Let $$Q(m)$$ be a nondegenerate polynomial and $$\Lambda$$ is corresponding set of regular values. Then for a test function $$\phi(x)\in\mathcal{S}(\mathbb{R}^n)$$ one has: $$\lim_{\lambda\in\Lambda,\lambda\to\infty}\frac{1}{r_Q(\lambda)}\sum_{Q(m)=\lambda}\phi(\lambda^{-1/d}m)=\int_{Q(x)=1}\phi(x)d\sigma_Q(x),$$ where $$r_Q(\lambda)= |\{m\in\mathbb{Z}^n : Q(m) = \lambda\}|$$ is the number of solutions $$m\in\mathbb{Z}^n$$ of the diophantine equation $$Q(m)=\lambda$$.

That is, when the solution sets $$Q(m)=\lambda$$ are projected to the unit surface $$Q(x)=1$$ via the dilations $$m\to\lambda^{-1/d}m$$, they weakly converge to the surface measure $$\frac{dS_Q(x)}{|Q'(x)|}$$. This is well-known in the case $$Q(x)$$ is a quadratic form. My request is: Can anyone indicate me some references containing this well-known result and its proof in the case $$Q(x)$$ is a quadratic form?

It is a whole line of ideas (and proofs) which go usually by the name of Linnik's problems''. Apart from Linnik's book (and the Linnik-Skubenko theorem), it has been extensively studied by many researchers, obviously Duke's original work and the Duke - Schulze-Pillot paper comes to mind.

If you assume high enough dimension, it mainly boils down to the Hardy-Littlewood method (Linnik was primarily concerned with the case of ternary forms, where this technique is not applicable).