# Uniform dimension results for fractional Brownian motion

### Richárd Balka

University of Washington, Seattle, USA and Alfréd Rényi Institute, Budapest, Hungary### Yuval Peres

Microsoft Research, Redmond, USA

## Abstract

Kaufman's dimension doubling theorem states that for a planar Brownian motion ${B(t):t∈[0,1]}$ we have

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 ${B(t):t∈[0,1]}$ be a fractional Brownian motion of Hurst index $α$. For a deterministic set $D⊂[0,1]$ consider the following statements:

(A) $P(dim_{H}B(A)=(1/α)dim_{H}Afor allA⊂D)=1$,

(B) $P(dim_{P}B(A)=(1/α)dim_{P}Afor allA⊂D)=1$,

(C) $P(dim_{P}B(A)≥(1/α)dim_{H}Afor allA⊂D)=1$.

We introduce a new concept of dimension, the modified Assouad dimension, denoted by $dim_{MA}$. We prove that $dim_{MA}D≤α$ 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 $dim_{MA}D≤α$. Furthermore, if $D$ is a set defined by digit restrictions then (A) holds if and only if $dim_{MA}D≤α$ or $dim_{H}D=0$. The characterization of (A) remains open in general. We prove that $dim_{MA}D≤α$ implies (B) and they are equivalent provided that $D$ is analytic. Let $D$ be compact, we show that (C) is equivalent to $dim_{H}D≤α$. This implies that if $dim_{H}D≤α$ and $Γ_{D}={E⊂B(D):dim_{H}E=dim_{P}E}$, then $P(dim_{H}(B_{−1}(E)∩D)=αdim_{H}Efor allE∈Γ_{D})=1.$ In particular, all level sets of $B∣_{D}$ have Hausdorff dimension zero almost surely.

## Cite this article

Richárd Balka, Yuval Peres, Uniform dimension results for fractional Brownian motion. J. Fractal Geom. 4 (2017), no. 2, pp. 147–183

DOI 10.4171/JFG/48