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The orbit SL(2, ℤ) x  is dense in ℝ 2  when the initial point  x ∈ℝ 2  has irrational slope. We refine this result from a diophantine perspective. For any target point  y ∈ℝ 2 , we introduce two exponents μ( x ,  y ) and μ̂( x ,  y ) that measure the approximation to  y  by elements γ  x  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 .

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