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Limit distributions of orbits of unipotent flows and values of quadratic forms. (English) Zbl 0814.22003
Gelfand, Sergej (ed.) et al., I. M. Gelfand Seminar. Part 1: Papers of the Gelfand seminar in functional analysis held at Moscow University, Russia, September 1993. Providence, RI: American Mathematical Society. Adv. Sov. Math. 16(1), 91-137 (1993).
The Oppenheim conjecture, proved by Margulis, is the following claim: Let $$Q$$ be an indefinite real quadratic form in at least three variables, which is not a scalar multiple of a rational form, and let $$I$$ be an interval in $$\mathbb{R}$$ of positive length. Then there exist integral $$n$$- tuples $$x$$ such that $$Q(x) \in I$$. In the paper under review the authors give a lower bound for the number of such solutions in a Euclidean ball of radius $$r$$ centered at 0 for all large $$r$$. The proof of the Oppenheim conjecture is based on the study of unipotent flows on homogeneous spaces $$G/\Gamma$$ where $$G$$ is a Lie group and $$\Gamma$$ is a lattice in $$G$$. A unipotent flow is the natural action of a unipotent one parameter subgroup $$U$$ of $$G$$ on $$G/\Gamma$$. By a result of Ratner the orbit $$Ux$$ of such a flow is uniformly distributed in its closure, which by a conjecture due to Raghunathan and proved by Ratner is a homogeneous space itself. The authors obtain results about how this uniform distribution depends on the initial point $$x$$. The number theoretic result mentioned above is a corollary.
For the entire collection see [Zbl 0777.00035].

##### MSC:
 22E40 Discrete subgroups of Lie groups 37C10 Dynamics induced by flows and semiflows 11H55 Quadratic forms (reduction theory, extreme forms, etc.) 37A99 Ergodic theory