Restrictions of H\"older continuous functions

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 $1\mathrm{/}2\le V(\alpha )\le 1\mathrm{/}(2-\alpha )$ and asked whether the upper bound is sharp. We show that in fact $V(\alpha )=\max \{1\mathrm{/}2,\alpha \}$. Let ${\dim }_H$ and ${\stackrel{}{\dim }}_M$ 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 $A\subset [0,1]$suchthat$\overline{\dim}_{M} A>\max{1-\alpha,\alpha}$and$B\colon A\to $\mathbb{R}$$ is of bounded variation. Furthermore, almost surely, there exists no set $A\subset [0,1]$ such that ${\stackrel{}{\dim }}_{M}A>1-\alpha$ and $B\text{\hspace{0.17em}⁣}:A\to \mathbb{R}$ 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 ${\dim }_{H}D\ge 1\mathrm{/}3$ and $\mathbf{B}\text{\hspace{0.17em}⁣}:D\to {\mathbb{R}}^2$ is non-decreasing in each coordinate. It remains open whether $1\mathrm{/}3$ is best possible.