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

Abstract

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 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.

On homogeneous and inhomogeneous Diophantine approximation over the fields of formal power series

“…Ghosh, Gorodnik, and Nevo [5,6] have studied the generic rate of approximation by lattice orbits for a large class of lattice actions on homogeneous varieties of connected almost simple, semisimple algebraic groups. Laurent and Nogueira [10] confined their investigations to the standard linear action of the lattice SL(2, Z) on the punctured plane R 2 \ {0}. In a previous work [16], the second-named author extended their approach and showed similar results for a few lattices inside SL(2, C) acting linearly on C 2 \ {0}.…”

“…The argument here is same as the one used in [10,16] except that we get tighter bounds owing to the ultrametric inequality. Now if k is chosen so as to have | Q k | ≤ | γ | < | Q k+1 |, we immediately get µ(x, 0) ≤ 1.…”

“…Our next goal is to show that the inequalities (4.19) and (4.21) are actually equalities when the target y has slope in K. We start with µ(x, y). The trick is to break down any matrix γ using the various convergent matrices for ξ and y, which are already familiar to us from (4.3) (see also [10,Theorem 4]). The theorem given below is an inhomogeneous version of Lemma 4.1 for rational target points.…”

“…Observe that τ is in the range [1/3, 1/2] as ω(y) ∈ [1, ∞]. Much like [10], our choice of the indices j and k in the construction of γ is governed by…”

Classifying orbits of the affine group over the integers

“…While the study of the orbits of the groups GL(2, Z), SL(2, Z), SL(2, Z) + requires techniques from various mathematical areas [1,2,4,8,9,10], a main possible reason of interest in our classification stems from the pervasive and novel role played by rational polyhedral geometry [3], through the fundamental notion of a (Farey) regular simplex [6]. Regular simplexes in R n are the affine counterparts of regular cones in Z n+1 .…”