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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: [email protected])
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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.

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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