Restrictions of Brownian motion

Abstract

Let $\{ B(t) \colon 0\leq t\leq 1\}$ be a linear Brownian motion and let $\dim$ denote the Hausdorff dimension. Let $α>\frac12$ and $1\leq β\leq 2$. We prove that, almost surely, there exists no set $A\subset[0,1]$ such that $\dim A>\frac12$ and $B\colon A\to\mathbb{R}$ is $α$-Hölder continuous. The proof is an application of Kaufman's dimension doubling theorem. As a corollary of the above theorem, we show that, almost surely, there exists no set $A\subset[0,1]$ such that $\dim A>\fracβ{2}$ and $B\colon A\to\mathbb{R}$ has finite $β$-variation. The zero set of $B$ and a deterministic construction witness that the above theorems give the optimal dimensions.