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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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References:
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