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Some effective results for ×a×b

Published online by Cambridge University Press:  21 May 2009

JEAN BOURGAIN ,
ELON LINDENSTRAUSS ,
PHILIPPE MICHEL  and
AKSHAY VENKATESH
JEAN BOURGAIN
Affiliation:
Institute for Advanced Study, Princeton, NJ 08540, USA
ELON LINDENSTRAUSS
Affiliation:
Princeton University, Princeton, NJ 08544, USA The Hebrew University, 91904 Jerusalem, Israel
PHILIPPE MICHEL
Affiliation:
EPFL, 1015 Lausanne, Switzerland Universit Montpellier, 34095 Montpellier, France
AKSHAY VENKATESH
Affiliation:
Stanford University, Stanford, CA 94305, USA Courant Institute of Mathematical Sciences, New York, NY 10012, USA
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Abstract

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We provide effective versions of theorems of Furstenberg and Rudolph–Johnson regarding closed subsets and probability measures of ℝ/ℤ invariant under the action of a non-lacunary multiplicative semigroup of integers. In particular, we give an explicit rate at which the sequence {anbkx}n,k becomes dense for a,b fixed multiplicatively independent integers and x∈ℝ/ℤ Diophantine generic.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 6 , December 2009 , pp. 1705 - 1722
Copyright
Copyright © Cambridge University Press 2009

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