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

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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ {\bold R}^n $\end{document} have Minkowski dimension at least \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ {n+2 \over 2} + \varepsilon_n $\end{document} for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ n \geq 4 $\end{document}, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ \varepsilon_n > 0 $\end{document} 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ n \ge 9 $\end{document}. Unlike the arguments in [KLT], [B3], [KT], our arguments will be purely geometric and do not require arithmetic combinatorics.