Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-25T06:06:05.813Z Has data issue: false hasContentIssue false
  • English
  • Français

Homotopy and the Kestelman–Borwein–Ditor Theorem

Published online by Cambridge University Press:  20 November 2018

N. H. Bingham  and
A. J. Ostaszewski
N. H. Bingham
Affiliation:
Mathematics Department, Imperial College London, South Kensington, London SW7 2AZ, U.K.e-mail: [email protected]
A. J. Ostaszewski
Affiliation:
Mathematics Department, London School of Economics, Houghton Street, London WC2A 2AE, U.K.e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

26A03measurecategorymeasure-category dualitydifferentiable homotopy
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 54 , Issue 1 , 01 March 2011 , pp. 12 - 20
Copyright
Copyright © Canadian Mathematical Society 2011

References

[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.Google Scholar
[BOst9] Bingham, N. H. and Ostaszewski, A. J., Infinite combinatorics in function spaces: category methods. Publ. Inst. Math. (Beograd) (N.S.) 86(100)(2009), 55–73.Google Scholar
[BOst11] Bingham, N. H. and Ostaszewski, A. J., Beyond Lebesgue and Baire II. Bitopology and measure-category duality. Colloq. Math., to appear.Google Scholar
[BOst12] Bingham, N. H. and Ostaszewski, A. J., Normed versus topological groups: dichotomy and duality. Dissertationes Math., to appear.Google Scholar
[BoDi] Borwein, D. and Ditor, S. Z., Translates of sequences in sets of positive measure. Canad. Math. Bull. 21(1978), no. 4, 497–498.Google Scholar
[Kes] Kestelman, H., The convergent sequences belonging to a set. J. London Math. Soc. 22(1947), 130–136. doi:10.1112/jlms/s1-22.2.130Google Scholar
[Komj-1] Komjáth, P., Large sets not containing images of a given sequence. Canad.Math. Bull. 26(1983), no. 1, 41–43.Google Scholar
[Komj-2] Komjáth, P., Large small sets. Colloq. Math. 56(1988), no. 2, 231–233.Google Scholar
[McSh] McShane, E. J., Images of sets satisfying the condition of Baire. Ann. of Math. (2) 51(1950), 380–386. doi:10.2307/1969330Google Scholar
[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. 415–431. http://www.math.wisc.edu/\symbol﹛126﹜miller/res/ Google Scholar
[MilH] Miller, H. I., Generalization of a result of Borwein and Ditor. Proc. Amer. Math. Soc. 105(1989), no. 4, 889–893. doi:10.2307/2047048Google Scholar
[Trau] Trautner, R., A covering principle in real analysis. Quart. J. Math. Oxford (2) 38(1987), no. 149, 127–130. doi:10.1093/qmath/38.1.127Google Scholar