Generalization of a result of Borwein and Ditor
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- by Harry I. Miller PDF
- Proc. Amer. Math. Soc. 105 (1989), 889-893 Request permission
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
- 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, DOI 10.4153/CMB-1978-084-5
- 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
- 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
Additional Information
- © Copyright 1989 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 105 (1989), 889-893
- MSC: Primary 28A05; Secondary 26A21
- DOI: https://doi.org/10.1090/S0002-9939-1989-0989098-5
- MathSciNet review: 989098