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?


1 Answer 1


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).


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