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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].

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