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

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.