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Generalization of a result of Borwein and Ditor


Author: Harry I. Miller
Journal: Proc. Amer. Math. Soc. 105 (1989), 889-893
MSC: Primary 28A05; Secondary 26A21
MathSciNet review: 989098
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Abstract: D. Borwein and S. Z. Ditor have found a measurable subset $ A$ of the real line having positive Lebesgue measure and a decreasing sequence $ \left( {{d_n}} \right)$ of reals converging to zero such that, for each $ x,x + {d_n}$ is not in $ A$ for infinitely many $ n$; thus answering a question of P. Erdös. It will be shown that the result of Borwein and Ditor can be extended using a general $ 2$-place function in place of plus. Related material is also presented.


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

  • [1] D. Borwein and S. Z. Ditor, Translates of sequences in sets of positive measure, Canad. Math. Bull. 21 (1978), no. 4, 497–498. MR 523593, 10.4153/CMB-1978-084-5
  • [2] Harry I. Miller, On certain transformations of sets, Akad. Nauka Umjet. Bosne Hercegov. Rad. Odjelj. Prirod. Mat. Nauka 24 (1985), 5–9 (English, with Serbo-Croatian summary). MR 837045
  • [3] John C. Oxtoby, Measure and category, 2nd ed., Graduate Texts in Mathematics, vol. 2, Springer-Verlag, New York-Berlin, 1980. A survey of the analogies between topological and measure spaces. MR 584443

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DOI: https://doi.org/10.1090/S0002-9939-1989-0989098-5
Article copyright: © Copyright 1989 American Mathematical Society