Packing-Dimension Profiles and Fractional Brownian Motion

In order to compute the packing dimension of orthogonal projections Falconer and Howroyd (1997) introduced a family of packing dimension profiles ${\mathrm{D}\mathrm{i}\mathrm{m}}_s$ 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.