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Values of isotropic quadratic forms at \(S\)-integral points. (English) Zbl 0777.11008
This article is concerned with a generalization of Margulis’ theorem that a real indefinite quadratic form in at least 3 variables that is not proportional to a rational form assumes arbitrarily small values at integral points. Let \(k\) be a number field, \(S\) a finite set of places including the archimedean places and \(k_ S\) the product of the \(k_ v\) for \(v\in S\). It is proved that for a nondegenerate indefinite quadratic form \(F\) on \(k^ n_ S\) that is not \(k_ S\)-proportional to a form on \(k^ n\) and \(n\geq 3\) there is an \(S\)-integral point \(x\) such that \(F(x)\) has arbitrary small nonzero absolute value at all \(v\in S\). In the case that the restriction of \(F\) to \(k_ v\) is \(k_ v\)-proportional to a form on \(k^ n\) for at least one of the valuations \(v\in S\) the proof proceeds in a way quite different from Margulis’ proof.
In an appendix it is shown with the help of recent results of Ratner that under the above assumptions the set of values of \(F\) is dense in \(k_ S\) if \(S\) is just the set of all archimedean places. An announcement of the results of this paper was given earlier [C. R. Acad. Sci., Paris, Ser. I 307, 217-220 (1988; Zbl 0654.10022)].

MSC:
11D75 Diophantine inequalities
11E12 Quadratic forms over global rings and fields
22E40 Discrete subgroups of Lie groups
11H50 Minima of forms
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