Let $q$ be a non-degenerate indefinite quadratic form over $ \mathbb{R}$
in $n \ge 3$ variables. Establishing a longstanding conjecture of Oppenheim, Margulis proved in 1986 that if $q$ is not a multiple of a rational form, then the set of values $q( \mathbb{Z}^n)$ is a dense subset of $ \mathbb{R}$.
Quantifying this result, Eskin, Margulis, and Mozes proved in 1986 that unless $q$ has signature $(2,1)$ or $(2,2)$, then the number $N(a,b;r)$ of integral vectors $v$ of norm at most $r$ satisfying $q(v) \in (a,b)$ has the asymptotic behavior $N(a,b;r) \sim \lambda(q) \cdot (b-a) r^{n-2}$.
Now, let $S$ is a finite set of places of $ \mathbb{Q}$ containing the Archimedean one, and $q=(q_v)_{v \in S}$
is an $S$-tuple of irrational isotropic quadratic forms over the completions $ \mathbb{Q}_v$. In this talk I will discuss the question of distribution of values of $q(v)$ as $v$ runes over $S$-balls in $ \mathbb{Z}[1/S]$.
This talk is based on a joint work with Seonhee Lim and Jiyoung Han.