On orbits of sl(2,_z )+ and values of binary quadratic forms on positive integral pairs

Abstract

We consider actions of SL(2, Z) and SL(2, Z)+ (semigroup of matrices with nonnegative integral entries) on the projective space P and on P×P. Results are obtained on orbit-closures under these actions and they are applied to describe a class of binary quadratic forms Q such that the sets Q(Z ²) or Q(Z +²) are dense in R. We prove also a result generalising a theorem of Troessaert and Valette (On values at integer points of some irrational quadratic forms, preprint) associating to certain binary quadratic forms Q rational quadratic forms Q' such that the closure of Q(Z ²) contains Q'(Z ²).