Effective estimates on indefinite ternary forms
type: Item
instance of: scholarly article
title: Effective estimates on indefinite ternary forms
zbMATH Open document ID: 1308.37002
author: Elon Lindenstrauss
author: Gregory Margulis
published in: Israel Journal of Mathematics
publication date: 2014-12-02T00:00:00Z
review text: Let \(Q\) be a quadratic form in three variables, with real coefficients. If \(Q\) is neither a multiple of a rational form nor definite (over \(\mathbb R\)) then a theorem of Dani and Margulis states that the set of the values of \(Q\) on the integers points in \({\mathbb Z}^3\) is dense in \(\mathbb R\) (previous results of Margulis implied that it has 0 as an accumulation point, solving a conjecture of Oppenheim). Given such a \(Q\), a real number \(\xi\) and \(\delta>0\) we thus know that there is a vector \(v\in{\mathbb Z}^3\) such that \(|\xi - Q(v)| < \delta\). The question that this paper adresses is that of determining such a \(v\) with the smallest possible Euclidean norm. The answer they give is the following: fixing \(\varepsilon > 0\), for any large enough \(T\) (depending on \(Q,\varepsilon\)) then: {\parindent=6mm \begin{itemize}\item[--] either we have \(\|Q-\lambda^{-1} Q'\| \leq \|Q\| T^{-1}\) where \(Q'\) is a form with coefficients in \(\mathbb Z\) and \(1 \leq \lambda \leq T^{\varepsilon/3}\), \item[--] or for every \(\xi\in\mathbb R\) with \(|\xi|\leq \log(T)^\kappa\) there is a \(v\in{\mathbb Z}^3,\, \|v\| \leq T^K\), such that \(|\xi - Q(v)| \leq KT^{-\kappa}\), \end{itemize}} where \(K\) is a positive absolute constant and \(\kappa = K^{-1}\). If we suppose in addition that \(Q\) has algebraic coefficients then the first possibility above cannot occur and it follows that (asymptotically) the answer to the question is \(\|v\| \leq e^{K\max(|\delta|,|\xi|)}\). The author notes that the constants in their results are effectively computable from their proof. The proof of the result follows the dynamical route initiated by Margulis, and continued in many works including that of \textit{S. G. Dani} and \textit{G. A. Margulis} [Invent. Math. 98, No. 2, 405--424 (1989; Zbl 0682.22008)] (we note that the introduction to the present paper recalls many facts about the history of the Oppenheim Conjecture). The basic observation is this: If \(Q\) is a qu…
reviewed by: Jean Raimbault
Mathematics Subject Classification ID: 37A17
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 22F30
Mathematics Subject Classification ID: 11E25
zbMATH DE Number: 6375692
zbMATH Keywords: Oppenheim conjecture
zbMATH Keywords: orbit closure
MaRDI profile type: Publication
full work available at URL: s11856-014-1110-3
OpenAlex ID: W1978489738
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DOI: 10.1007/S11856-014-1110-3
description: scientific article; zbMATH DE number 6375692