Restrictions of Hölder continuous functions

For $0<α<1$ let $V(α)$ denote the supremum of the numbers $v$ such that every $α$-Hölder continuous function is of bounded variation on a set of Hausdorff dimension $v$. Kahane and Katznelson (2009) proved the estimate $1/2 \leq V(α)\leq 1/(2-α)$ and asked whether the upper bound is sharp. We show that in fact $V(α)=\max\{1/2,α\}$. Let $\dim_{H}$ and $\overline{\dim}_{M}$ denote the Hausdorff and upper Minkowski dimension, respectively. The upper bound on $V(α)$ is a consequence of the following theorem. Let $\{B(t): t\in [0,1]\}$ be a fractional Brownian motion of Hurst index $α$. Then, almost surely, there exists no set $A\subset [0,1]$ such that $\overline{\dim}_{M} A>\max\{1-α,α\}$ and $B\colon A\to \mathbb{R}$ is of bounded variation. Furthermore, almost surely, there exists no set $A\subset [0,1]$ such that $\overline{\dim}_{M} A>1-α$ and $B\colon A\to \mathbb{R}$ is $β$-Hölder continuous for some $β>α$. 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 $α$-Hölder 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.

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