An improved bound for the Minkowski dimension of Besicovitch sets in medium dimension

Details An improved bound for the Minkowski dimension of Besicovitch sets in medium dimension An improved bound for the Minkowski dimension of Besicovitch sets in medium dimension

2000-04-04

arxiv.org/abs/math/0004015v1

Mathematics

Classical Analysis and ODEs

31 pages, 3 figures, submitted GAFA

Scientific paper

We use geometrical combinatorics arguments, including the ``hairbrush'' and x-ray arguments of Wolff and the sticky/plany/grainy analysis of Katz, Laba, and Tao, to show that Besicovitch sets in R^n have Minkowski dimension at least (n+2)/2 + \eps_n for all n > 3, where \eps_n > 0 is an absolute constant depending only on n. This complements the results of Katz, Laba, and Tao, which established the same result for n=3, and of Bourgain and Katz-Tao, arithmetic combinatorics techniques to establish the result for n > 8. In contrast to previous work, our arguments will be purely geometric and do not require arithmetic combinatorics.

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