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

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 $n\geq 4$ 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.