A generalization of Furstenberg's

Diophantine Theorem

Author:
Bryna Kra

Journal:
Proc. Amer. Math. Soc. **127** (1999), 1951-1956

MSC (1991):
Primary 11J71, 54H20

Published electronically:
February 18, 1999

MathSciNet review:
1487320

Full-text PDF Free Access

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.**Daniel Berend,*Multi-invariant sets on tori*, Trans. Amer. Math. Soc.**280**(1983), no. 2, 509–532. MR**716835**, 10.1090/S0002-9947-1983-0716835-6**2.**Daniel Berend,*Actions of sets of integers on irrationals*, Acta Arith.**48**(1987), no. 3, 275–306. MR**921090****3.**D. Berend and M. D. Boshernitzan,*Numbers with complicated decimal expansions*, Acta Math. Hungar.**66**(1995), no. 1-2, 113–126. MR**1313779**, 10.1007/BF01874357**4.**Michael D. Boshernitzan,*Elementary proof of Furstenberg’s Diophantine result*, Proc. Amer. Math. Soc.**122**(1994), no. 1, 67–70. MR**1195714**, 10.1090/S0002-9939-1994-1195714-X**5.**Harry Furstenberg,*Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation*, Math. Systems Theory**1**(1967), 1–49. MR**0213508****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.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
11J71,
54H20

Retrieve articles in all journals with MSC (1991): 11J71, 54H20

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:
http://dx.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