Approximation to points in the plane by SL (2, Z)-orbits
Résumé
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]Origine | Fichiers produits par l'(les) auteur(s) |
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