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 $0<α<1$ and let $\{B(t): t\in [0,1]\}$ be a fractional Brownian motion of Hurst index $α$. For a deterministic set $D\subset [0,1]$ consider the following statements: $$(A) \quad \mathbb{P}(\dim_H B(A)=(1/α) \dim_H A \textrm{ for all } A\subset D)=1,$$ $$(B) \quad \mathbb{P}(\dim_P B(A)=(1/α) \dim_P A \textrm{ for all } A\subset D)=1, $$ $$(C) \quad \mathbb{P}(\dim_P B(A)\geq (1/α) \dim_H A \textrm{ for all } A\subset D)=1.$$ We introduce a new concept of dimension, the modified Assouad dimension, denoted by $\dim_{MA}$. We prove that $\dim_{MA} D\leq α$ implies (A), which enables us to reprove a restriction theorem of Angel, Balka, Máthé, and Peres. We show that if $D$ is self-similar then (A) is equivalent to $\dim_{MA} D\leq α$. Furthermore, if $D$ is a set defined by digit restrictions then (A) holds iff $\dim_{MA} D\leq α$ or $\dim_H D=0$. The characterization of (A) remains open in general. We prove that $\dim_{MA} D\leq α$ implies (B) and they are equivalent provided that $D$ is analytic. We show that (C) is equivalent to $\dim_H D\leq α$. This implies that if $\dim_H D\leq α$ and $Γ_D=\{E\subset B(D): \dim_H E=\dim_P E\}$, then $$\mathbb{P}(\dim_H (B^{-1}(E)\cap D)=α\dim_H E \textrm{ for all } E\in Γ_D)=1.$$ In particular, all level sets of $B|_{D}$ have Hausdorff dimension zero almost surely.

27 pages. Lemma 4.3 in the earlier version was incorrect, so we removed it and generalized Lemma 4.4, see the new Lemma 4.3. The published paper only states Theorem 1.9 for compact sets