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

## TL;DR

This paper extends the quantitative Oppenheim conjecture results to $S$-arithmetic quadratic forms of rank 3 and 4, showing that almost all such forms satisfy the conjecture despite exceptions in low dimensions.

## Contribution

It generalizes the quantitative Oppenheim conjecture to the $S$-arithmetic setting for quadratic forms of rank 3 and 4, including almost all forms.

## Key findings

- Almost all $S$-arithmetic quadratic forms of rank 3 and 4 satisfy the conjecture.
- Extension of previous results from real to $S$-arithmetic quadratic forms.
- The result holds despite known exceptions in low-dimensional cases.

## Abstract

The celebrated result of Eskin, Margulis and Mozes (1998) and Dani and Margulis (1993) on quantitative Oppenheim conjecture says that for irrational quadratic forms $q$ of rank at least 5, the number of integral vectors $\mathbf v$ such that $q(\mathbf v)$ is in a given bounded interval is asymptotically equal to the volume of the set of real vectors $\mathbf v$ such that $q(\mathbf v)$ is in the same interval. In dimension $3$ or $4$, there are exceptional quadratic forms which fail to satisfy the quantitative Oppenheim conjecture. Even in those cases, one can say that 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 result to the $S$-arithmetic version.