Uniform dimension results for fractional Brownian motion

Abstract

Kaufman's dimension doubling theorem states that for a planar Brownian motion $\{\mathbf{B}(t): t\in [0,1]\}$ we have $$\mathbb{P}(\dim \mathbf{B}(A)=2\dim A \textrm{ for all } A\subset [0,1])=1,$$ where $\dim$ may denote both Hausdorff dimension $\dim_H$ and packing dimension $\dim_P$. The main goal of the paper is to prove similar uniform dimension results in the one-dimensional case. Let $0<\alpha<1$ and let $\{B(t): t\in [0,1]\}$ be a fractional Brownian motion of Hurst index $\alpha$. For a deterministic set $D\subset [0,1]$ consider the following statements: $$(A) \quad \mathbb{P}(\dim_H B(A)=(1/\alpha) \dim_H A \textrm{ for all } A\subset D)=1,$$ $$(B) \quad \mathbb{P}(\dim_P B(A)=(1/\alpha) \dim_P A \textrm{ for all } A\subset D)=1, $$ $$(C) \quad \mathbb{P}(\dim_P B(A)\geq (1/\alpha) \dim_H A \textrm{ for all } A\subset D)=1.$$ We introduce a new concept of dimension, the modified Assouad dimension, denoted by $\dim_{MA}$. We prove that $\dim_{MA} D\leq \alpha$ implies (A), which enables us to reprove a restriction theorem of Angel, Balka, M\'ath\'e, and Peres. We show that if $D$ is self-similar then (A) is equivalent to $\dim_{MA} D\leq \alpha$. Furthermore, if $D$ is a set defined by digit restrictions then (A) holds iff $\dim_{MA} D\leq \alpha$ or $\dim_H D=0$. The characterization of (A) remains open in general. We prove that $\dim_{MA} D\leq \alpha$ implies (B) and they are equivalent provided that $D$ is analytic. We show that (C) is equivalent to $\dim_H D\leq \alpha$. This implies that if $\dim_H D\leq \alpha$ and $\Gamma_D=\{E\subset B(D): \dim_H E=\dim_P E\}$, then $$\mathbb{P}(\dim_H (B^{-1}(E)\cap D)=\alpha \dim_H E \textrm{ for all } E\in \Gamma_D)=1.$$ In particular, all level sets of $B|_{D}$ have Hausdorff dimension zero almost surely.