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Homotopy and the Kestelman–Borwein–Ditor Theorem

  • N. H. Bingham (a1) and A. J. Ostaszewski (a2)
Abstract

The Kestelman–Borwein–Ditor Theorem, on embedding a null sequence by translation in (measure/category) “large” sets has two generalizations. Miller replaces the translated sequence by a “sequence homotopic to the identity”. The authors, in a previous paper, replace points by functions: a uniform functional null sequence replaces the null sequence, and translation receives a functional form. We give a unified approach to results of this kind. In particular, we show that (i) Miller's homotopy version follows fromthe functional version, and (ii) the pointwise instance of the functional version follows from Miller's homotopy version.

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References
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[BGT] Bingham, N. H., Goldie, C. M., and Teugels, J. L., Regular variation. Encyclopedia of Mathematics and its Applications, 27, Cambridge University Press, Cambridge, 1989.
[BOst9] Bingham, N. H. and Ostaszewski, A. J., Infinite combinatorics in function spaces: category methods. Publ. Inst. Math. (Beograd) (N.S.) 86(100)(2009), 5573.
[BOst11] Bingham, N. H. and Ostaszewski, A. J., Beyond Lebesgue and Baire II. Bitopology and measure-category duality. Colloq. Math., to appear.
[BOst12] Bingham, N. H. and Ostaszewski, A. J., Normed versus topological groups: dichotomy and duality. Dissertationes Math., to appear.
[BoDi] Borwein, D. and Ditor, S. Z., Translates of sequences in sets of positive measure. Canad. Math. Bull. 21(1978), no. 4, 497498.
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[Mil] A. W.|Miller, Special sets of reals. In: Set theory of the reals, Israel Math. Conf. Proc., 6, Bar-Ilan Univ., Ramat Gan, 1993, pp. 415431. http://www.math.wisc.edu/\symbol﹛126﹜miller/res/
[MilH] Miller, H. I., Generalization of a result of Borwein and Ditor. Proc. Amer. Math. Soc. 105(1989), no. 4, 889893. doi:10.2307/2047048
[Trau] Trautner, R., A covering principle in real analysis. Quart. J. Math. Oxford (2) 38(1987), no. 149, 127130. doi:10.1093/qmath/38.1.127
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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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