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

Michel Laurent (1) , Arnaldo Nogueira (1)

1 I2M - Institut de Mathématiques de Marseille

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 µ(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 µ(x, y) in terms of the irrationality measure of the slope of x.

Domaines: Mathématiques [math]

Michel Laurent, Arnaldo Nogueira. Approximation to points in the plane by SL (2, Z)-orbits. Journal of the London Mathematical Society, 2012, ⟨10.1112/jlms/jdr061⟩. ⟨hal-01262176⟩