Vol 4, 1994

INVARIANT MEASURES AND ORBIT CLOSURES

245

Margulis did. Subsequently he and Dani [DM2] showed that the values of В at the primitive elements of Z^ are dense in R. Recently A. Borel and G. Prasad [BoP] have obtained a remarkable strengthening of this fact, implied by Theorem 4.

THEOREM 02 (Borel, Prasad [BoP]). Let В be as in Theorem 01. Then given С1,...,Сгг-1 G R and e > О there are xi,.. .^Xn-i E Z'^ which are part of a basis in Z^ (and hence are primitive elements of Z^) such that \В{хг) — Сг\ < 6 for aii г = 1,... ,n — 1.

Theorems 01 and 02 are proved in the next section.

In fact, A. Borel and G. Prasad [BoP] have generalized the Oppenheim Conjecture and Theorem 02 to the following more general setting.

Let fc be a number field and о the ring of integers of k. For every normalized absolute value | • 1^ on fc, let ky be the completion of к at v. Let 5 be a finite set of places of к containing the set Soo of archimedean ones, к s the direct sum of the fields kg^ s £ S and о s the ring of 5-integers of к (i.e. of elements x ^ к with |a:|^ < 1 for all v ^ 5).

A quadratic form F on k^ is a collection (F^), s G 5, where Fg is a quadratic form on k^. The form is nondegenerate if and only if each Fg is nondegenerate. The form is isotropic if each Fg is so, i.e. if there exists for each s E 5 an element Xg E k^ — {0} such that Fs{xs) = 0. If s is a real place, this condition is equivalent to Fg being indefinite. The form F is said to be rational if there exists a unit A = (A^) E kg and a form Fq on k^ such that Fg = XgFo for all 5 G 5, irrational otherwise.

THEOREM 03 (Borel, Prasad [BoP, Theorem A]). Let F be irrational, nondegenerate, isotropic and n > 3. Tien given s > 0 there exists x E o^ such that 0 < \Fg{x)\ < e for all s e S.

THEOREM 04 (Borel, Prasad [BoP, Corollary 7.9]). Assume S = S^ stnd let F be as in Theorem 03. Let Ai,...,An-i G ks- Then for each j = 1,2,... tliere are х^д,..., Xj^n-i E:0^ = Og which are part of a basis of o" over 0 (and hence are primitive elements ofo^) such that lim^_oo F[xj^^) = A^ for aii г = 1,..., n ~ 1. In particular, the set of values of F on the primitive elements of o^ is dense in ks.

Theorems 03 and 04 in [BoP] are deduced by means of Theorem 4, geometry of numbers and strong approximation in algebraic groups. The density of F{o^) for nonarchimedean S follows (see [BoP]) from the 5- arithmetic version of Theorem 4 (see [Ra7, Theorem 8]).

Finally we mention the following problem. Let Л be a quadratic form as in Theorem 01. Given 0 < a < 6 and r > 0, let Er{a,b) = {a; G Z"" : a < \B{x)\ < b, ||a;|| < r}. Then card Er{a,b) -^ oo when r -^ oo. It seems