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Loi des grands nombres pour des sommes de Riesz–Raikov multidimensionnelles

Published online by Cambridge University Press:  04 December 2007

EMMANUEL LESIGNE
EMMANUEL LESIGNE
Affiliation:
Départment de Mathématiques, Faculté des Sciences et Techniques, Université François Rabelais, parc de Grandmont, 37200 Tours, France e-mail: [email protected]
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Abstract

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Let A and x be d × d and d × 1 real matrices. We study the asymptotic distribution of the sequence ((A$^n$x) mod.Z$^d$) in the torus $T^d$. We prove that for any A, for almost all x, this distribution exists; we characterize the case where this distribution is uniform; we give a description of the non uniform case. Finally we ask a question on the asymptotic distribution modulo 1 of the coefficients of the sequence of powers (A$^n$).

Keywords

distribution modulo 1uniform distributionmultidimensional distributionweyl criteriakoksma metric theorem.
Type
Research Article
Information
Compositio Mathematica , Volume 110 , Issue 1 , January 1998 , pp. 39 - 49
Copyright
© 1998 Kluwer Academic Publishers