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 and are dense in the circle for all irrational , where are distinct pairs of multiplicatively independent integers for .

**1.**D. Berend. Multi invariant set on tori.*Trans. Amer. Math. Soc.*, 280:509-532, 1983. MR**85b:11064****2.**D. Berend. Actions of sets of integers on irrationals.*Acta. Arith.*, 48:175-306, 1987. MR**89a:11077****3.**D. Berend and M. D. Boshernitzan. Numbers with complicated decimal expansions.*Acta Math. Hungar.*, 66(1-2):113-126, 1995. MR**95m:11073****4.**M.D. Boshernitzan. Elementary proof of Furstenberg's diophantine result.*Proc. Amer. Math. Soc.*, 122(1):67-70, 1994. MR**94k:11085****5.**H. Furstenberg. Disjointness in ergodic theory, minimal sets, and a problem in diophantine approximation.*Math. Systems Theory*, 1:1-49, 1967. MR**35:4369****6.**G.H. Hardy and J.E. Littlewood. The fractional part of .*Acta. Math.*, 37:155-191, 1914.**7.**D. Meiri. Entropy and uniform distribution of orbits in .*to appear, Isr. J. Math.*, 1998.

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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