Asymptotic distribution of values of isotropic quadratic forms at $S$-integral points

Abstract

We prove an analogue of a theorem of Eskin-Margulis-Mozes: suppose we are given a finite set of places $S$ over $\mathbb{Q}$ containing the archimedean place and excluding the prime $2$, an irrational isotropic form $\mathbf{q}$ of rank $n\ge 4$ on ${\mathbb{Q}}_S$, a product of $p$-adic intervals $I_p$, and a product $\Omega$ of star-shaped sets. We show that unless $n=4$ and $\mathbf{q}$ is split in at least one place, the number of $S$-integral vectors $\mathbf{v}\in \mathsf{T}\Omega$ satisfying simultaneously $\mathbf{q}(\mathbf{v})\in I_p$ for $p\in S$ is asymptotically given by $$\lambda (\mathbf{q},\Omega )|I|\cdot \prod _{p\in S_f}{T}_p^{n-2},$$ as $\mathsf{T}$ goes to infinity, where $|I|$ is the product of Haar measures of the $p$-adic intervals $I_p$. The proof uses dynamics of unipotent flows on $S$-arithmetic homogeneous spaces; in particular, it relies on an equidistribution result for certain translates of orbits applied to test functions with a controlled growth at infinity, specified by an $S$-arithmetic variant of the $\alpha$-function introduced in the work of Eskin, Margulis, Mozes, and an $S$-arithemtic version of a theorem of Dani-Margulis.

arXiv commentThe article will appear in Journal of Modern Dynamics. In the revised version, several typos have been fixed. Moreover, another preprint (arXiv:1605.02436) has been incorporated

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