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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
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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] R. Raimi, Translation properties of finite partitions of the positive integers, Fund. Math. 61 (1968), 253-256. MR 0222874 (36:5924)
  • [2] C. Ryll-Nardzewski, Remark on Raimi's theorem on translations, Fund. Math. 61 (1968), 257-258. MR 0222875 (36:5925)
  • [3] N. Hindman, Ultrafilters and combinatorial number theory, Lecture Notes in Math., Vol. 751, Springer-Verlag, Berlin, 1979. MR 564927 (81m:10019)
  • [4] D. G. Champernowne, The construction of decimals normal in the scale of ten, J. London Math. Soc. 8 (1933), 254-260.
  • [5] R. von Mises, Über Zahlenfolgen die ein Kollektiv-ahnliches Verhalten zeigen, Math. Ann. 108 (1933), 757-772. MR 1512874
  • [6] H. Davenport and P. Erdös, Note on normal decimals, Canad. J. Math. 4 (1952), 58-63. MR 0047084 (13:825g)

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

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