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

TL;DR

This paper studies how well points in the plane with irrational slopes can be approximated by the orbits of SL(2,Z) acting on a given point, providing effective bounds based on the size of the group elements.

Contribution

It offers effective approximation results for points in the plane by SL(2,Z)-orbits, extending understanding of orbit density with quantitative bounds.

Findings

Provides explicit bounds on approximation quality.

Shows the density of SL(2,Z)-orbits is effective and quantifiable.

Extends classical density results with effective approximation rates.

Abstract

Let x be a point in R^2 with irrational slope and let \Gamma denote the lattice SL(2,Z) acting linearly on R^2. Then, the orbit \Gamma x is dense in R^2. We give efective results on the approximation of a point y in R^2 by points of the form \gamma x, where \gamma belongs to \Gamma, in terms of the size of \gamma.