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Lattice orbit distribution on ℝ2

Published online by Cambridge University Press:  21 July 2009

ARNALDO NOGUEIRA
ARNALDO NOGUEIRA*
Affiliation:
Institut de Mathématiques de Luminy, 163, avenue de Luminy, Case 907, 13288 Marseille Cedex 9, France (email: nogueira@iml.univ-mrs.fr)
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Abstract

We study the distribution on ℝ2 of the orbit of a vector under the linear action of SL(2,ℤ). Let Ω⊂ℝ2 be a compact set and x∈ℝ2. Let N(k,x) be the number of matrices γ∈SL(2,ℤ) such that γ(x)∈Ω and ‖γ‖≤k, k=1,2,…. If Ω is a square, we prove the existence of an absolute error term for N(k,x), as k→∞, for almost every x, which depends on the Diophantine property of the ratio of the coordinates of x. Our approach translates the question into a Diophantine approximation counting problem which provides the absolute error term. The asymptotical behaviour of N(k,x) is also obtained using ergodic theory.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 4 , August 2010 , pp. 1201 - 1214
Copyright
Copyright © Cambridge University Press 2009

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