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Relatively prime numbers and invariant measures under the natural action of SL (n, \mathbb{Z}) on \mathbb{R}^{n}

Published online by Cambridge University Press:  19 June 2002

ARNALDO NOGUEIRA
ARNALDO NOGUEIRA
Affiliation:
Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, 21945-970 Rio de Janeiro, RJ, Brazil
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Abstract

The linear action of SL (n, \mathbb{Z}) on \mathbb{R}^n has non-trivial invariant sets and matrices of SL (n, \mathbb{Z}) preserve volume. Here we prove that a Borel measure on \mathbb{R}^n, locally finite at a point, invariant under the linear action of SL (n, \mathbb{Z}) annihilating the set \{tz: t\geq 0,z\in \mathbb{Z}^n\} must be a scalar multiple of the Lebesgue measure. Our result extends a theorem due to S. G. Dani concerning locally finite measures, ergodic under the linear action of SL (n, \mathbb{Z}). Our approach is alternative, it exploits the Euclidean structure of the space and the fact that the probability of n integers being relatively prime is strictly positive.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 22 , Issue 3 , June 2002 , pp. 899 - 923
Copyright
2002 Cambridge University Press

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