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How many intervals cover a point in Dvoretzky covering?

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Abstract

Consider the Dvoretzky random covering with length sequence {α/n} n≥1 (α>0). We are interested in the setF β of points on the circle which are covered by a numberβ logn of the firstn randomly placed intervals. It is proved among others that for a certain interval ofβ>0, the Hausdorff dimension ofF β is equal to 1−[βlog(β/α)−(β−α)]. This implies that points on the circle are differently covered.

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Authors and Affiliations

  1. Department of Mathematics, Wuhan University, 430072, Wuhan, China

    Aihua Fan

  2. Département de Mathématiques et Informatique, Université de Picardie Jules Verne, 33, Rue Saint Leu, 80039, Amiens Cedex 1, France

    Aihua Fan

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  1. Aihua Fan
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Correspondence to Aihua Fan.

Additional information

The research was partially supported by the Zheng Ge Ru Foundation and the RGC grant of Hong Kong.

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Fan, A. How many intervals cover a point in Dvoretzky covering?. Isr. J. Math. 131, 157–184 (2002). https://doi.org/10.1007/BF02785856

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  • DOI: https://doi.org/10.1007/BF02785856

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