Restrictions of Hölder continuous functions

Abstract: For $0<\alpha<1$ let $V(\alpha)$ denote the supremum of the numbers $v$ such that every $\alpha$-H\"older continuous function is of bounded variation on a set of Hausdorff dimension $v$. Kahane and Katznelson (2009) proved the estimate $1/2 \leq V(\alpha)\leq 1/(2-\alpha)$ and asked whether the upper bound is sharp. We show that in fact $V(\alpha)=\max{1/2,\alpha}$. Let $\dim_{H}$ and $\overline{\dim}{M}$ denote the Hausdorff and upper Minkowski dimension, respectively. The upper bound on $V(\alpha)$ is a consequence of the following theorem. Let ${B(t): t\in [0,1]}$ be a fractional Brownian motion of Hurst index $\alpha$. Then, almost surely, there exists no set $A\subset [0,1]$ such that $\overline{\dim}{M} A>\max{1-\alpha,\alpha}$ and $B\colon A\to \mathbb{R}$ is of bounded variation. Furthermore, almost surely, there exists no set $A\subset [0,1]$ such that $\overline{\dim}{M} A>1-\alpha$ and $B\colon A\to \mathbb{R}$ is $\beta$-H\"older continuous for some $\beta>\alpha$. The zero set and the set of record times of $B$ witness that the above theorems give the optimal dimensions. We also prove similar restriction theorems for deterministic self-affine functions and generic $\alpha$-H\"older continuous functions. Finally, let ${\mathbf{B}(t): t\in [0,1]}$ be a two-dimensional Brownian motion. We prove that, almost surely, there is a compact set $D\subset [0,1]$ such that $\dim{H} D\geq 1/3$ and $\mathbf{B}\colon D\to \mathbb{R}^2$ is non-decreasing in each coordinate. It remains open whether $1/3$ is best possible.