A generalization of Furstenberg’s Diophantine Theorem
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- by Bryna Kra PDF
- Proc. Amer. Math. Soc. 127 (1999), 1951-1956 Request permission
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 $i=1, 2$.References
- Daniel Berend, Multi-invariant sets on tori, Trans. Amer. Math. Soc. 280 (1983), no. 2, 509–532. MR 716835, DOI 10.1090/S0002-9947-1983-0716835-6
- Daniel Berend, Actions of sets of integers on irrationals, Acta Arith. 48 (1987), no. 3, 275–306. MR 921090, DOI 10.4064/aa-48-3-275-306
- D. Berend and M. D. Boshernitzan, Numbers with complicated decimal expansions, Acta Math. Hungar. 66 (1995), no. 1-2, 113–126. MR 1313779, DOI 10.1007/BF01874357
- Michael D. Boshernitzan, Elementary proof of Furstenberg’s Diophantine result, Proc. Amer. Math. Soc. 122 (1994), no. 1, 67–70. MR 1195714, DOI 10.1090/S0002-9939-1994-1195714-X
- Harry Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory 1 (1967), 1–49. MR 213508, DOI 10.1007/BF01692494
- G.H. Hardy and J.E. Littlewood. The fractional part of $n^k\theta$. Acta. Math., 37:155–191, 1914.
- D. Meiri. Entropy and uniform distribution of orbits in ${T}^d$. to appear, Isr. J. Math., 1998.
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
- MR Author ID: 363208
- ORCID: 0000-0002-5301-3839
- Email: bryna@math.lsa.umich.edu, kra@ihes.fr
- 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
- © Copyright 1999 American Mathematical Society
- 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
- MathSciNet review: 1487320