Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Elementary proof of Furstenberg's Diophantine result


Author: Michael D. Boshernitzan
Journal: Proc. Amer. Math. Soc. 122 (1994), 67-70
MSC: Primary 11K31
DOI: https://doi.org/10.1090/S0002-9939-1994-1195714-X
MathSciNet review: 1195714
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We present an elementary proof of a diophantine result (due to H. Furstenberg) which asserts (in a special case) that for every irrational $ \alpha $ the set $ \{ {2^m}{3^n}\alpha \vert m,n \geq 0\} $ is dense modulo 1. Furstenberg's original proof employs the theory of disjointness of topological dynamical systems.


References [Enhancements On Off] (What's this?)

  • [AHK] M. Ajtai, I. Havas, and J. Komlós, Every group admits a bad topology, Stud. Pure Math., Memory of P. Turan, Birkhäuser, Basel and Boston, 1983, pp. 21-34. MR 820207 (86k:22002)
  • [Be1] D. Berend, Multi-Invariant sets on tori, Trans. Amer. Math. Soc. 280 (1983), 509-532. MR 716835 (85b:11064)
  • [Be2] -, Multi-Invariant sets on compact abelian groups, Trans. Amer. Math. Soc. 286 (1984), 505-535. MR 760973 (86e:22009)
  • [Be3] -, Actions of sets of integers on irrationals, Acta Arith. 48 (1987), 275-306. MR 921090 (89a:11077)
  • [Be4] -, Dense $ \pmod 1$ dilated semigroups of algebraic numbers, J. Number Theory 26 (1987), 246-256. MR 901238 (88e:11102)
  • [Be5] -, private communication.
  • [Bos1] M. Boshernitzan, Homogeneously distributed sequences of integers of sublucunary growth, Monatsh. Math. 96 (1983), 173-181. MR 725629 (85k:11034)
  • [Bos2] -, Density modulo 1 of dilations of sublacunary sequences, Adv. Math. (to appear). MR 1293584 (95g:11005)
  • [Bou] J. Bourgain, On the maximal ergodic theorem for certain subsets of the integers, Israel J. Math. 61 (1988), 39-72. MR 937581 (89f:28037a)
  • [ET] P. Erdös and S. J. Taylor, On the set of points of convergence of a lacunary trigonometric series and the equidistribution properties of related sequences, Proc. London Math. Soc. (3) 7 (1957), 598-615. MR 0092032 (19:1050b)
  • [F] H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in diophantine approximation, Math. Systems Theory 1 (1967), 1-49. MR 0213508 (35:4369)
  • [M1] D. de Mathan, Sur un probleme de densite modulo 1, C. R. Acad. Sci. Paris Sér. I Math. 287 (1978), 277-279. MR 510723 (80a:10072)
  • [M2] -, Numbers contravening a condition in density modulo 1, Acta Math. Hungar. 36 (1980), 237-241. MR 612195 (82e:10088)
  • [P] A. D. Pollington, On the density of sequence $ \{ {n_k}\theta \} $, Illinois J. Math. 23 (1979), 151-155. MR 540398 (80i:10066)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 11K31

Retrieve articles in all journals with MSC: 11K31


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1994-1195714-X
Keywords: Distribution modulo 1, topological dynamics
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society