  We study the action of a lattice in the group SL(2,R) on the plane. We obtain
a formula which simultaneously describes visits of an orbit to either a fixed
ball, or an expanding or contracting family of annuli. We also discuss the
`shrinking target problem'. Our results are valid for an explicitly described
set of initial points: all nonzero vectors in the case of a cocompact lattice,
and all vectors satisfying certain diophantine conditions in case SL(2,Z). The
proofs combine the method of Ledrappier with effective equidistribution results
for the horocycle flow due to Burger, Strombergsson, Forni and Flaminio.
