Approximation to points in the plane by SL (2, ℤ)-orbits

The orbit SL(2,Z)x is dense in R2 when the initial point x ∈ R2 has irrational slope. We refine this result from a diophantine perspective. For any target point y ∈ R2, we introduce two exponents μ(x,y) and μ̂(x,y) that measure the approximation to y by elements γx of the orbit in terms of the size of γ. We estimate both exponents under various conditions. Our results are optimal when the slope of the target point y is a rational number. In that case we express μ(x,y) and μ̂(x,y) in terms of the irrationality measure of the slope of x.