An Improved Bound on the Minkowski Dimension of Besicovitch Sets in ℝ 3

Abstract

Abstract. We use geometrical combinatorics arguments, including the “hairbrush” argument of Wolff [11], the x-ray estimates in [12], [7], and the sticky/plany/grainy analysis of [6], to show that Besicovitch sets in Rn have Minkowski dimension at least n+2 + εn for all n ≥ 4, where εn> 0 is an absolute constant 2 depending only on n. This complements the results of [6], which established the same result for n = 3, and of [3], [5], which used arithmetic combinatorics techniques to establish the result for n ≥ 9. Unlike the arguments in [6], [3], [5], our arguments will be purely geometric and do not require arithmetic combinatorics. 1.