an:06991111
Zbl 1407.11084
Han, Jiyoung; Lim, Seonhee; Mallahi-Karai, Keivan
Asymptotic distribution of values of isotropic here quadratic forms at \(S\)-integral points
EN
J. Mod. Dyn. 11, 501-550 (2017).
00416284
2017
j
11H50 22E40 22D40 11P21 05C15 37E25 68R15
Oppenheim conjecture; homogeneous dynamic
Summary: We prove an analogue of a theorem of Eskin-Margulis-Mozes [\textit{A. Eskin} et al., Ann. Math. (2) 147, No. 1, 93--141 (1998; Zbl 0906.11035)]. 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 \(\mathsf{I}_p\), 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
\[
\lambda({\mathbf q}, \Omega) | \mathsf{I}| \cdot \| {\mathsf{T}} \|^{n-2}
\]
as \({\mathsf{T}}\) goes to infinity, where \(|\mathsf{I}|\) 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 [loc. cit.], and an \(S\)-arithemtic version of a theorem of [\textit{S. G. Dani} and \textit{G. A. Margulis}, in: I. M. Gelfand Seminar. Part 1: Papers of the Gelfand seminar in functional analysis held at Moscow University, Russia, September 1993. Providence, RI: American Mathematical Society. 91--137 (1993; Zbl 0814.22003)].
Zbl 0906.11035; Zbl 0814.22003