Dimensional Properties of One-Dimensional Brownian Motion

Abstract

For each closed set $F \subseteq \lbrack 0, 1\rbrack, \dim X(F + t) = \min(1, 2 \dim F)$ for almost all $t > 0. (X$ is one-dimensional Brownian motion). For each closed set $F \subseteq \lbrack 0, 1 \rbrack$ of dimension greater than $1/2, m(X(F + t)) > 0$ for almost all $t > 0$. These statements are true outside a single null-set in the sample space.