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Lattice orbit distribution on ℝ2

Published online by Cambridge University Press:  21 July 2009

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)

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
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Dani, S. G.. On uniformly distributed orbits of certain horocycle flows. Ergod. Th. & Dynam. Sys. 2 (1982), 139158.CrossRefGoogle Scholar
[2]Estermann, T.. On the number of primitive lattice points in a parallelogram. Canad. J. Math. 5 (1953), 456459.Google Scholar
[3]Hardy, G. H. and Wright, E. M.. An Introduction to the Theory of Numbers, 3rd edn. Oxford University Press, Oxford, 1964.Google Scholar
[4]Huxley, M. N.. Area, Lattice Points and Exponential Sums, 1st edn(London Mathematical Society Monographs, New Series, 13). Oxford University Press, New York, 1996.CrossRefGoogle Scholar
[5]Kesten, H.. On a conjecture of Erdös and Szusz related to uniform distribution mod 1. Acta Arith. 12 (1966), 193212.Google Scholar
[6]Kuipers, L. and Niederreiter, H.. On Uniform Distribution of Sequences. Wiley, New York, 1974.Google Scholar
[7]Ledrappier, F.. Distribution des orbites des réseaux sur le plan réel. C. R. Math. Acad. Sci. Paris 329 (1999), 6164.Google Scholar
[8]Nogueira, A.. Orbit distribution on ℝ2 under the linear action of SL(2,ℤ). Indag. Math. (N.S.) 13(1) (2002), 103124.CrossRefGoogle Scholar
[9]Schoissengeier, J.. On the discrepancy of () II. J. Number Theory 24 (1986), 5464.CrossRefGoogle Scholar
[10]Schoissengeier, J.. The discrepancy of ()n≥1. Math. Ann. 296 (1993), 529545.CrossRefGoogle Scholar
[11]Watson, T.. Integers n relatively prime to [αn]. Canad. J. Math. 5 (1953), 451455.CrossRefGoogle Scholar