Restrictions of Brownian motion

Abstract

Let $\{B(t):0\le t\le 1\}$ be a linear Brownian motion and let $\dim$ denote the Hausdorff dimension. Let $\alpha >\frac{1}{2}$ and $1\le \beta \le 2$. We prove that, almost surely, there exists no set $A\subset [0,1]$ such that $\dim A>\frac{1}{2}$ and $B:A\to \mathbb{R}$ is $\alpha$-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 $\dim A>\frac{\beta }{2}$ and $B:A\to \mathbb{R}$ has finite $\beta$-variation. The zero set of $B$ and a deterministic construction witness that the above theorems give the optimal dimensions.

6 pages