Restrictions of H\"older continuous functions

TL;DR

This paper determines the exact maximum Hausdorff dimension for subsets of [0,1] on which certain stochastic and deterministic functions, including fractional Brownian motion and self-affine functions, exhibit bounded variation or Hölder continuity.

Contribution

It establishes the precise value of V(α) for H"older functions and proves optimal restriction theorems for fractional Brownian motion and other functions, resolving previous bounds.

Findings

V(α) = max{1/2, α} for H"older continuous functions.

Almost surely, fractional Brownian motion has no bounded variation on sets with Minkowski dimension exceeding max{1-α, α}.

Existence of a set D with Hausdorff dimension ≥ 1/3 where 2D Brownian motion is coordinate-wise non-decreasing.

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.