  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.
