Asymptotic distribution of values of isotropic here quadratic forms at S-integral points

Jiyoung Han; Seonhee Lim; Keivan Mallahi-Karai

doi:10.3934/jmd.2017020

Article Contents

2017, Volume 11: 501-550. Doi: 10.3934/jmd.2017020

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Asymptotic distribution of values of isotropic here quadratic forms at S-integral points

1.
Department of Mathematical Sciences, Seoul National University, Kwanak-ro 1, Kwanak-gu, Seoul 08826, Republic of Korea
2.
Department of Mathematics, Jacobs University, 28759 Bremen, Germany

Received: June 08, 2016

Revised: August 2017

Published: October 2017

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Abstract

We prove an analogue of a theorem of Eskin-Margulis-Mozes [10]. 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

$\begin{split}\lambda({\mathbf q}, \Omega) |\,\mathsf{I}\,| \cdot \| {\mathsf{T}} \|^{n-2}\end{split}$

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 [10], and an $S$ -arithemtic version of a theorem of Dani-Margulis [7].

Keywords:

Oppenheim conjecture,
homogeneous dynamics.

Mathematics Subject Classification: Primary: 20E08, 20F65; Secondary: 05C15, 37E25, 68R15.

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