Exceptional points for Lebesgue's density theorem on the real line |
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Authors: | András Szenes |
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Institution: | Section de Mathématiques, Université de Genève, 2-4 rue du Lièvre, 1211 Genève, Switzerland |
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Abstract: | For a nontrivial measurable set on the real line, there are always exceptional points, where the lower and upper densities of the set are neither 0 nor 1. We quantify this statement, following work by V. Kolyada, and obtain the unexpected result that there is always a point where the upper and the lower densities are closer to 1/2 than to zero or one. The method of proof uses a discretized restatement of the problem, and a self-similar construction. |
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Keywords: | Lebesgue density theorem Measurable sets Fractals Interval configurations |
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