Packing intermediate dimensions of measures and applications

Source abstract

We introduce a spectrum of dimensions of measures, the \emph{packing intermediate dimensions} $\dim_{P,θ}μ$, $θ\in(0,1]$, of a Borel probability measure $μ$ on $\mathbb{R}^n$ and the \emph{quasi-Hausdorff dimension} $\dim_{qH}μ$. We show that this spectrum encodes the information on $μ$ that governs the packing dimension of its projections and Hölder images. First, $\dim_{qH}μ$ is the exact threshold for the packing dimension profiles of Falconer and Howroyd: for every real $0<s\leq n$, $\dim_P^sμ=s$ if and only if $s\leq\dim_{qH}μ$. Second, we prove a lower bound for the profiles in terms of $\dim_Hμ$, $\dim_{P,θ}μ$ and the Assouad dimension of $\operatorname{supp}μ$, which improves the bound of Falconer and Mattila, the best previously known general lower bound for the packing dimension of projected measures. The improvement is strict whenever $\dim_Hμ<\dim_Pμ$ and $\dim_A(\operatorname{supp}μ)<n$, and when $\dim_A(\operatorname{supp}μ)=n$ it occurs exactly when $\dim_{P,θ}μ$ exceeds an explicit universal lower bound for some $θ$ in an explicit range. We present examples of measures with $\dim_Hμ<\dim_Pμ<\dim_A(\operatorname{supp}μ)=n$ for which our bound gives the exact value $\dim_Pμ_V=\min\{\dim_Pμ,m\}$ for almost every $m$-dimensional subspace $V$, whereas the bound of Falconer and Mattila is strictly smaller. As applications, for a large family of random $α$-Hölder maps $f_ω:\mathbb{R}^n\to\mathbb{R}^m$, we prove that, for compactly supported $μ$, $\dim_Pμ_{f_ω}=m$ almost surely if and only if $αm\leq\dim_{qH}μ$, and that the packing analogue of Kahane's formula holds for fractional Brownian motions if and only if $\dim_{qH}μ=\dim_Pμ$. Analogous results are obtained for analytic sets.