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

We use geometrical combinatorics arguments, including the "hair-brush" argu ment of Wolff [W1], the x-ray estimates in [W2], [LT], and the sticky/plany /grainy analysis of [KLT], to show that Besicovitch sets in R-n have Minkow ski dimension at least n+2/2 + epsilon (n) for all n greater than or equal to 4, where epsilon (n) > 0 is an absolute constant depending only on n. Th is 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 greater than or equal to 9. Unlike the argument s in [KLT], [B3], [KT], our arguments will be purely geometric and do not r equire arithmetic combinatorics.