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Some effective results for ×a×b

  • JEAN BOURGAIN (a1), ELON LINDENSTRAUSS (a2) (a3), PHILIPPE MICHEL (a4) (a5) and AKSHAY VENKATESH (a6) (a7)
Abstract

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.

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References
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[1]Baker, A. and Wüstholz, G.. Logarithmic forms and group varieties. J. Reine Angew. Math. 442 (1993), 1962.
[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.
[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.
[4]Einsiedler, M., Lindenstrauss, E., Michel, P. and Venkatesh, A.. The distribution of periodic torus orbits on homogeneous spaces Duke Math. J. to appear.
[5]Furstenberg, H.. Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Systems Theory 1 (1967), 149.
[6]Host, B.. Nombres normaux, entropie, translations. Israel J. Math. 91(1–3) (1995), 419428.
[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.
[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.
[9]Rudolph, D. J.. ×2 and ×3 invariant measures and entropy. Ergod. Th. & Dynam. Sys. 10(2) (1990), 395406.
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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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