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

Abstract

We use geometrical combinatorics arguments, including the hairbrush argument of Wolff [W1], the x-ray estimates in [W2], [LT], and the sticky/plany/grainy analysis of [KLT], to show that Besicovitch sets in \( {\bold R}^n \) have Minkowski dimension at least \( {n+2 \over 2} + \varepsilon_n \) for all \( n \geq 4 \), where \( \varepsilon_n > 0 \) is an absolute constant depending only on n. This complements the results of [KLT], which established the same result for n = 3, and of [B3], [KT], which used arithmetic combinatorics techniques to establish the result for \( n \ge 9 \). Unlike the arguments in [KLT], [B3], [KT], our arguments will be purely geometric and do not require arithmetic combinatorics.