Values of quadratic forms at primitive integral points
type: Item
instance of: scholarly article
title: Values of quadratic forms at primitive integral points
zbMATH Open document ID: 0682.22008
DOI: 10.1007/BF01388860
author: Gregory Margulis
author: S. G. Dani
published in: Inventiones Mathematicae
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143736
review text: The authors continue \textit{G. Margulis}' investigation of the set of values of real indefinite quadratic forms with the help of ergodic methods [Proc. Symp. in Honor of A. Selberg, Oslo 1987, 377--398 (1989; Zbl 0675.10010)]). Let \(B\) be such a nondegenerate form in at least 3 variables that is not proportional to a rational form and \(T\) a binary form represented by \(T\) over the reals. Then it is proved that there are binary forms arbitrary close to \(T\) that are represented by \(B\) with primitive integral representing vectors. (It would be interesting to know whether the representing pair of integral vectors can be chosen such that it spans a direct summand of \(\mathbb Z^n)\). The result is shown to depend on the fact that all orbits in \(\mathrm{SL}(3,\mathbb R)/\mathrm{SL}(3,\mathbb Z)\) of the connected component of 1 of the special orthogonal group of B are either closed or dense. The proof of this fact (and results on flows on homogeneous spaces of Lie groups necessary for it) occupy the main part of the paper.
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 11E12
Mathematics Subject Classification ID: 11H55
zbMATH DE Number: 4116840
zbMATH Keywords: values of real indefinite quadratic forms
zbMATH Keywords: primitive integral representing vectors
zbMATH Keywords: orbits
zbMATH Keywords: special orthogonal group
zbMATH Keywords: flows on homogeneous spaces of Lie groups
reviewed by: Rainer Schulze-Pillot
MaRDI profile type: Publication
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OpenAlex ID: W2104328103
description: scientific article; zbMATH DE number 4116840