Orbit distribution on [formula omitted]2 under the natural action of SL(2, [formula omitted

Abstract

Let x be a point in R 2. In the present paper, we give an explicit description of the set { Ax: A ϵ SL(2, Z )}, the orbit of x, under the natural action of SL(2, Z ) on R 2. Our approach allows us to analyze the distribution of an orbit in the real plane. As an application we evaluate, for r, t > 0, the asymptotic behaviour of the set {A∈SL(2,Z): Ax 0∈[0,r] 2, ‖A‖≦t} , where x 0=(( 5 −1) 2 ,1) and ‖A‖=max |a ij|, for A=(a ij) .