Uniform dimension results for fractional Brownian motion

TL;DR

This paper investigates uniform dimension results for fractional Brownian motion on subsets of [0,1], introducing the modified Assouad dimension to characterize when the Hausdorff and packing dimensions of images are scaled predictably.

Contribution

The paper introduces the modified Assouad dimension and establishes its role in uniform dimension results for fractional Brownian motion, extending previous theorems to one-dimensional cases.

Findings

Modified Assouad dimension characterizes dimension scaling for fractional Brownian motion.

Uniform dimension results hold for self-similar and digit-restricted sets under certain conditions.

All level sets of the fractional Brownian motion restricted to D have Hausdorff dimension zero almost surely.

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.