Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture
type: Item
instance of: scholarly article
title: Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture
zbMATH Open document ID: 0906.11035
DOI: 10.2307/120984
author: Alex Eskin
author: Shahar Mozes
author: Gregory Margulis
published in: Annals of Mathematics. Second Series
publication date: 1998-11-11T00:00:00Z
review text: Let \(Q\) be an indefinite nondegenerate quadratic form in \(n\) variables. \textit{G. Margulis} [Math. Ann. 286, 101-128 (1990; Zbl 0679.22007)] proved the Oppenheim conjecture, which states that if \(n\geq 3\) and \(Q\) is not proportional to a form with rational coefficients, then the set of values of \(Q\) at integral points is dense. Subsequently, using Ratner's measure classification theorem, as well as their own method of studying unipotent flows on homogeneous spaces via linearization, \textit{S. Dani} and \textit{G. Margulis} [Adv. Sov. Math. 16, 91-137 (1993; Zbl 0814.22003)] proved a refined version of Ratner's uniform distribution theorem. This theorem is the main ergodic result used in this paper. The main results of this paper are as follows: Let \(Q\) be an indefinite form of signature \((p,q)\) with \(p\geq 3\), \(q\geq 1\) and \(Q\) not proportional to a rational form. Then for any interval \((a,b)\) as \(T\to\infty\) \[ | V_{(a,b)}(\mathbb{Z})\cap T\Omega|\sim \lambda_{Q,\Omega} (b-a)T^{n-2}, \tag{1} \] where \(n=p+q\), \(\Omega= \{v\in\mathbb{R}^n\mid \| v\|< \rho(v/\| v\|)\}\), \(\rho\) is a continuous positive function on the unit sphere, \(T\Omega\) is the dilate of \(\Omega\) by \(T\), \(V_{(a,b)}(Z)= \{z\in\mathbb{Z}^n\mid a<Q(x)< b\}\) and \(\lambda_{Q,\Omega}\) satisfies \[ \text{Vol} (V_{(a,b)} (\mathbb{R})\cap T\Omega)\sim \lambda_{Q,\Omega} (b-a)T^{n-2}. \tag{2} \] Only the upper bound is new. If the signature of \(Q\) is \((2,1)\) or \((2,2)\), then no such universal formula exists. However an upper bound of the form \(T^q\log T\) is obtained. This upper bound is effective and is uniform over compact sets in the set of quadratic forms. An effective uniform upper bound of the form \(T^{n-2}\) for the case \(p\geq 3\) is also obtained. It is also proved that for almost all quadratic forms \(Q\) of signature \((p,q)= (2,1)\) or \((2,2)\) the asymptotic relation (1) holds. Finally, let \(p\geq 3\), \(q\geq 1\), \(D\) be a compact subset of …
Mathematics Subject Classification ID: 11H50
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 22D40
Mathematics Subject Classification ID: 11P21
zbMATH DE Number: 1192409
zbMATH Keywords: unipotent flows
zbMATH Keywords: Haar measure
zbMATH Keywords: ergodic theory on discrete groups
zbMATH Keywords: Oppenheim conjecture
zbMATH Keywords: compact sets
zbMATH Keywords: quadratic forms
zbMATH Keywords: effective uniform upper bound
zbMATH Keywords: asymptotic relation
Wikidata QID: Q123189817
reviewed by: Ranjeet Sehmi
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/120984
OpenAlex ID: W1974099924
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description: scientific article; zbMATH DE number 1192409