Acta Mathematica Hungarica

, Volume 134, Issue 3, pp 209–268 | Cite as

Optimal quality of exceptional points for the Lebesgue density theorem

Article

Abstract

In spite of the Lebesgue density theorem, there is a positive δ such that, for every non-trivial measurable set S⊂ℝ, there is a point at which both the lower densities of S and of ℝ∖S are at least δ. The problem of determining the supremum of possible values of this δ was studied in a paper of V. I. Kolyada, as well as in some recent papers. We solve this problem in the present work.

Key words and phrases

Lebesgue density theorem exceptional point interval configuration 

2010 Mathematics Subject Classification

28A99 

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References

  1. [1]
    M. Csörnyei, J. Grahl and T. C. O’Neil, Points of middle density in the real line, preprint, available e.g. at http://www.homepages.ucl.ac.uk/~ucahmcs/publ/index.html.
  2. [2]
    V. I. Kolyada, On the metric Darboux property, Analysis Math., 9 (1983), 291–312. CrossRefMATHMathSciNetGoogle Scholar
  3. [3]
    A. Szenes, Exceptional points for Lebesgue’s density theorem on the real line, Adv. Math., 226 (2011), 764–778. CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2011

Authors and Affiliations

  1. 1.Department of Mathematical Analysis, Faculty of Mathematics and PhysicsCharles UniversityPrague 8Czech Republic

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