On Orbits of SL(2,$\text{Z}$)₊ and Values of Binary Quadratic Forms on Positive Integral Pairs

Abstract

We consider actions of SL(2,$\text{Z}$) and SL(2,$\text{Z}$)₊ (semigroup of matrices with nonnegative integral entries) on the projective space $\text{P}$ and on $\text{P}$×$\text{P}$. 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($\text{Z}$²) or Q($\text{Z}$₊²) are dense in $\text{R}$. 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($\text{Z}$²) contains Q′($\text{Z}$²).