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arXiv · 1206.5645

Sur un ensemble de Besicovitch

Jean-Pierre Kahane
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Abstract

Here is an example of a plane set of vanishing area and consisting of line-segments whose directions cover an angle : let E be a Cantor set of dissection ratio 1/4 (therefore dimension 1/2) carried by the horizontal axis and E' the image of E by an homothetic transformation of ratio 2 whose center is not on the horizontal axis ; the union of the line segments joining E and E' is the set in question. The example was given in 1969 by the author with a wrong proof. The present article contains a short proof based on the projection theorem of Besicovitch, and a long proof resulting from the investigation of the arithmetic, geometric and analytic properties of the horizontal sections of the set. This investigation copies the study by Richard Kenyon of a similar problem. Variations of the construction are provided in the plane and in multidimensional spaces.

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Jean-Pierre Kahane. 2012-06-25. Sur un ensemble de Besicovitch. https://arxiv.org/abs/1206.5645

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