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.
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References
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.
V. I. Kolyada, On the metric Darboux property, Analysis Math., 9 (1983), 291–312.
A. Szenes, Exceptional points for Lebesgue’s density theorem on the real line, Adv. Math., 226 (2011), 764–778.
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This research was supported in part by the grant GAČR 201/09/0067 and in part by the grant SVV-2010-261316.
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Kurka, O. Optimal quality of exceptional points for the Lebesgue density theorem. Acta Math Hung 134, 209–268 (2012). https://doi.org/10.1007/s10474-011-0182-3
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DOI: https://doi.org/10.1007/s10474-011-0182-3