Packing-Dimension Profiles and Fractional Brownian Motion

Abstract

In order to compute the packing dimension of orthogonal projections Falconer and Howroyd (1997) introduced a family of packing dimension profiles ${\mathrm{Dim}}_s$ that are parametrized by real numbers $s>0$. Subsequently, Howroyd (2001) introduced alternate $s$-dimensional packing dimension profiles $\hbox{${\rm P}$-$\dim$}_s$ and proved, among many other things, that $\hbox{${\rm P}$-$\dim$}_s E={\rm Dim}_s E$ for all integers $s>0$ and all analytic sets $E\subseteq {\mathbb{R}}^N$. The goal of this article is to prove that $\hbox{${\rm P}$-$\dim$}_s E={\rm Dim}_s E$ for all real numbers $s>0$ and analytic sets $E\subseteq {\mathbb{R}}^N$. This answers a question of Howroyd (2001, p. 159). Our proof hinges on a new property of fractional Brownian motion.

References in corpus (1)

• A uniform result for the dimension of fractional Brownian motion level sets

Cited by in corpus (2)

• Hausdorff and packing dimensions of the images of random fields
• On dynamical bit sequences