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Some effective results for ×a×b

Published online by Cambridge University Press:  21 May 2009

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
Copyright
Copyright © Cambridge University Press 2009

References

[1]Baker, A. and Wüstholz, G.. Logarithmic forms and group varieties. J. Reine Angew. Math. 442 (1993), 1962.Google Scholar
[2]Bourgain, J., Glibichuk, A. and Konyagin, S.. Estimate for the number of sums and products and for exponential sums in fields of prime order. J. London Math. Soc. 73 (2006), 380398.CrossRefGoogle Scholar
[3]Bourgain, J.. Sum-product theorems and exponential sum bounds in residue classes for general modulus. C. R. Math. Acad. Sci. Paris 344(6) (2007), 349352.Google Scholar
[4]Einsiedler, M., Lindenstrauss, E., Michel, P. and Venkatesh, A.. The distribution of periodic torus orbits on homogeneous spaces Duke Math. J. to appear.Google Scholar
[5]Furstenberg, H.. Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Systems Theory 1 (1967), 149.CrossRefGoogle Scholar
[6]Host, B.. Nombres normaux, entropie, translations. Israel J. Math. 91(1–3) (1995), 419428.CrossRefGoogle Scholar
[7]Johnson, A. S. A.. Measures on the circle invariant under multiplication by a nonlacunary subsemigroup of the integers. Israel J. Math. 77(1–2) (1992), 211240.CrossRefGoogle Scholar
[8]Parry, W.. Squaring and cubing the circle—Rudolph’s theorem. Ergodic Theory of Z d Actions (Warwick, 1993–1994) (London Mathematical Society Lecture Note Series, 228). Cambridge University Press, Cambridge, 1996, pp. 177183.Google Scholar
[9]Rudolph, D. J.. ×2 and ×3 invariant measures and entropy. Ergod. Th. & Dynam. Sys. 10(2) (1990), 395406.CrossRefGoogle Scholar