Upper Bounds and Asymptotics in a Quantitative Version of the Oppenheim Conjecture

Quadratic forms of signature (2,2) and eigenvalue spacings on rectangular 2-tori

“…Recall that in [EMM,Th. 2.3] we proved that for Q of signature (p, q) with p ≥ 3 and q ≥ 1, and as long as Q is not proportional to a rational form (so that SO(Q)Λ is not closed), and for any continuous function ν on K, we have…”

“…This was used to obtain the main result [EMM,Th. 2.1] (which is restated as Theorem 1.1 in the present paper).…”

“…Theorem 1.4 follows from Theorem 2.3 by an argument identical to that used in [EMM,§3.4,§3.5] (9)), with the natural modifications due to the fact that we assert and assume only inequalities. Essentially, the proof is based on the identity of the form…”

Spherical averages in the space of marked lattices

“…Hence sets D 0 = A 0 × B 0 where the closure of A 0 has small Lebesgue measure have small probability, and we can thus remove such sets from the D i . This explains why, without loss of generality, we may assume from now on that the A i are convex and that the hyperplane 8) and write q = m(1, …”

“…If the space X is homogeneous, then the weak convergence of the spherical average P t can be proved for all x 0 , with a complete classification of all limit measures, by means of measure rigidity techniques that are based on Ratner's measure classification theorem for subgroups generated by unipotent elements [22]. There is now a large body of literature on this topic, see for instance [7,8,16,17,19,24]. In some settings, spherical equidistribution may also be deduced directly from the mixing property of R [7].…”