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)



Translation properties of sets of positive upper density

Authors: Vitaly Bergelson and Benjamin Weiss
Journal: Proc. Amer. Math. Soc. 94 (1985), 371-376
MSC: Primary 11K16
MathSciNet review: 787875
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Generalizing a result of Raimi we show that there exists a set $ E \subset {\mathbf{N}}$ such that if $ A \subset {\mathbf{N}}$ is a set with positive upper density, then there exists a number $ k \in {\mathbf{N}}$ such that $ {d^ * }((A + k) \cap E) > 0$ and $ {d^ * }((A + k) \cap {E^c}) > 0$. Some extensions and further results are also obtained.

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

  • [1] Ralph A. Raimi, Translation properties of finite partitions of the positive integers, Fund. Math. 61 (1967/1968), 253–256. MR 0222874
  • [2] C. Ryll-Nardzewski, Remark on Raimi’s theorem on translations, Fund. Math. 61 (1967/1968), 257–258. MR 0222875
  • [3] Neil Hindman, Ultrafilters and combinatorial number theory, Number theory, Carbondale 1979 (Proc. Southern Illinois Conf., Southern Illinois Univ., Carbondale, Ill., 1979) Lecture Notes in Math., vol. 751, Springer, Berlin, 1979, pp. 119–184. MR 564927
  • [4] D. G. Champernowne, The construction of decimals normal in the scale of ten, J. London Math. Soc. 8 (1933), 254-260.
  • [5] R. v. Mises, Über Zahlenfolgen, die ein kollektiv-ähnliches Verhalten zeigen, Math. Ann. 108 (1933), no. 1, 757–772 (German). MR 1512874,
  • [6] H. Davenport and P. Erdös, Note on normal decimals, Canadian J. Math. 4 (1952), 58–63. MR 0047084

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 11K16

Retrieve articles in all journals with MSC: 11K16

Additional Information

Keywords: Density, normal number
Article copyright: © Copyright 1985 American Mathematical Society