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A generalization of Furstenberg's
Diophantine Theorem


Author: Bryna Kra
Journal: Proc. Amer. Math. Soc. 127 (1999), 1951-1956
MSC (1991): Primary 11J71, 54H20
DOI: https://doi.org/10.1090/S0002-9939-99-04742-5
Published electronically: February 18, 1999
MathSciNet review: 1487320
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Abstract | References | Similar Articles | Additional Information

Abstract: We obtain a generalization of Furstenberg's Diophantine Theorem on non-lacunary multiplicative semigroups. For example we show that the sets of sums $\{(p_1^nq_1^m + p_2^nq_2^m)\alpha:n,m \in \mathbb{N}\}$ and $\{(p_1^nq_1^m + 2^n)\alpha: n,m \in \mathbb{N}\}$ are dense in the circle $\mathbb{T} = \mathbb{R}/ \mathbb{Z}$ for all irrational $\alpha$, where $(p_i, q_i)$ are distinct pairs of multiplicatively independent integers for $i=1, 2$.


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Additional Information

Bryna Kra
Affiliation: Department of Mathematics, University of Michigan, East Hall, 525 East University Avenue, Ann Arbor, Michigan 49108-1109
Address at time of publication: IHES, 35, route de Chartres, 91440 Bures-sur-Yvette, France
Email: bryna@math.lsa.umich.edu, kra@ihes.fr

DOI: https://doi.org/10.1090/S0002-9939-99-04742-5
Keywords: Topological dynamics, distribution modulo $1$
Received by editor(s): March 19, 1997
Received by editor(s) in revised form: October 2, 1997
Published electronically: February 18, 1999
Communicated by: Mary Rees
Article copyright: © Copyright 1999 American Mathematical Society

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