A BRIEF REMARK ON ORBITS OF SL(2,Z) IN THE EUCLIDEAN PLANE

A striking phenomenon is the gaps around lines of simple rational slopes. This appears for any initial point. We will describe here these gaps in a fully elementary way for the lattice Γ = SL(2,Z) (which is enough to describe it for all arithmetic lattices). Let us mention that our analysis is carried on in the arithmetic case for sake of elementariness but a similar analysis can be done for non-arithmetic lattices. Another experimentation with a cocompact lattice Λ does not show these gaps. It will be clear from the analysis below that this comes from the unique ergodicity of the unipotent flow in Λ\SL(2,R).