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

Abstract

The celebrated result of Eskin, Margulis and Mozes [8] and Dani and Margulis [7] on quantitative Oppenheim conjecture says that for irrational quadratic forms q of rank at least 5, the number of integral vectors v such that q(v) is in a given bounded interval is asymptotically equal to the volume of the set of real vectors v such that q(v) is in the same interval. In rank 3 or 4, there are exceptional quadratic forms which fail to satisfy the quantitative Oppenheim conjecture. Even in those cases, one can say that two asymptotic limits coincide for almost all quadratic forms ([8, Theorem 2.4]). In this paper, we extend this result to the S-arithmetic version.

Values of inhomogeneous forms at S ‐integral points

“…In [17], Han et al obtained 𝑆-arithmetic generalizations of the quantitative Oppenheim problem in rank 5 and higher. The case of forms of rank 3 and 4 was studied by Han in [18]. Effective results for generic (homogeneous) forms in the 𝑆-arithmetic setting were obtained by Han in [16], who also established an 𝑆-arithmetic version of Rogers' mean value formula.…”

Values of Inhomogeneous Forms at S-integral points

“…In [16], Han, Lim and Mallahi-Karai obtained S-arithmetic generalizations of the quantitative Oppenheim problem in rank 5 and higher. The case of forms of rank 3 and 4 was studied by Han in [18]. Effective results for generic (homogeneous) forms in the S-arithmetic setting were obtained by Han in [17], who also established an S-arithmetic version of Rogers' mean value formula.…”