Quantitative version of the Oppenheim conjecture for inhomogeneous quadratic forms
type: Item
instance of: scholarly article
title: Quantitative version of the Oppenheim conjecture for inhomogeneous quadratic forms
zbMATH Open document ID: 1243.11054
DOI: 10.1215/00127094-1276319
author: Amir Mohammadi
author: Gregory Margulis
published in: Duke Mathematical Journal
publication date: 2011-06-06T00:00:00Z
full work available at URL: https://arxiv.org/abs/1001.2756
review text: Let \(Q\) be a non-degenerate indefinite quadratic form on \(\mathbb R^n\). Let \(\xi\in\mathbb R^n\) be a vector, and define the (inhomogeneous) quadratic form \(Q_\xi\) by \(Q_\xi= Q(x+\xi)\) for all \(x\in\mathbb R^n\). Let \(v\) be a continuous function on the sphere \(\{v\in\mathbb R^n:\| v\|= 1\}\). Define \(\Omega= \{v\in\mathbb R^n:\| v\|< \nu(v/\| v\|\}\), and let \(T\Omega\) be the dilate of \(\Omega\) by \(T\). Let \(N_{Q,\xi,\Omega}= \#\{x\in\mathbb Z^n: x\in T\Omega\) and \(a< Q_\xi(x)< b\}\). If \(\xi= 0\), we let \(N_{Q,\xi,\Omega}(a,b,T)= N_{Q,\Omega}(a,b, T)\). It is easy to see that there exists a constant \(\lambda_{Q,\Omega}\) such that \(\text{Vol}(\{x\in\mathbb R^n: x\in T\Omega\) and \(a< Q_\xi(x)< b\})\sim\lambda_{Q,\xi}(b- a)T^{n-2}\). \textit{A. Eskin}, the first author and \textit{S. Mozes} [Electron. Res. Announc. Am. Math. Soc. 1, No. 3, 124--130 (1995; Zbl 0859.11043); Ann. Math. (2) 147, No. 1, 93--141 (1998; Zbl 0906.11035); Ann. Math. (2) 161, No. 2, 679--725 (2005; Zbl 1075.11049)] proved that if \(Q\) be a quadratic form not proportional to a rational form and of signature \((p,q)\) with \(p\geq 3\) and \(q\geq 1\), then for any interval \((a,b)\), we have \[ N_{Q,\Omega}(a,b,T)\sim \lambda_{Q,\xi}(b- a)T^{n-2} \] as \(T\to\infty\), where \(n= p+q\). If \(Q\) is an irrational quadratic form of signature \((2,2)\) which is not extremely well approximable by split forms (EWAS) then \[ \widetilde N_{Q,\Omega}(a, b, T)\sim\lambda_{Q,\xi}(b- a)T^2 \] as \(T\to\infty\), where \(\widetilde N_{Q, \xi, \Omega}(a,b,T)= \#\{x\in\mathbb Z^n: x\) is not a null subspace of \(Q_\xi\), \(x\in T\Omega\) and \(a< Q_\xi(x)< b\}\). This paper extends the above results to inhomogeneous quadratic forms as follows: 1. Let \(Q_\xi\) be an inhomogeneous quadratic form of signature \((2,2)\). Assume that \(Q_\xi\) is Diophantine. Then for any interval \((a,b)\), we have \[ \widetilde N_{Q,\xi,\Omega}(a,b,T)\sim \lambda_{Q,\Omega}(b-a)T^2 \] as \(T\to\infty\…
reviewed by: Ranjeet Sehmi
Mathematics Subject Classification ID: 11E20
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 5904313
zbMATH Keywords: Oppenheim conjecture
zbMATH Keywords: quadratic form
zbMATH Keywords: Diophantine
Wikidata QID: Q123242526
MaRDI profile type: Publication
OpenAlex ID: W2964141379
arXiv ID: 1001.2756
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description: scientific article