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The celebrated result of Eskin, Margulis and Mozes (1998) and Dani andMargulis (1993) on quantitative Oppenheim conjecture says that for irrationalquadratic forms $q$ of rank at least 5, the number of integral vectors $\mathbfv$ such that $q(\mathbf v)$ is in a given bounded interval is asymptoticallyequal to the volume of the set of real vectors $\mathbf v$ such that $q(\mathbfv)$ is in the same interval.  In dimension $3$ or $4$, there are exceptional quadratic forms which fail tosatisfy the quantitative Oppenheim conjecture. Even in those cases, one can saythat almost all quadratic forms hold that two asymptotic limits are the same([Eskin-Margulis-Mozes'98, Theorem 2.4]). In this paper, we extend this resultto the $S$-arithmetic version.

Quantitative Oppenheim conjecture for $S$-arithmetic quadratic forms of rank $3$ and $4$