Two-Scale Frostman Measures
Nicolas Angelini, Ursula Molter
TL;DR
The paper addresses unifying Frostman-type measure characterizations of fractal dimensions with the $\\dim_\\theta$-intermediate dimensions. It introduces the dyadic dimension $\\mathcal{D}(E)$ and proves the existence of a family of Radon measures $\\{\\mu_\\delta\\}$ supported on $E$ that satisfy a two-scale Frostman-type decay: $\\mu_\\delta(B(x,r)) \\le c (\\delta^{1/\\theta})^{t-s} r^{s}$ for $r<\\delta^{1/\\theta}$ and $\\mu_\\delta(B(x,r)) \\le c r^{t}$ for $\\delta^{1/\\theta} \\le r \\le \\delta$, with $0< t < \\overline{\\dim}_{\\theta} E$ and $0< s < \\mathcal{D}(E)$. This yields a unified, two-scale dimension characterization linking $\\dim_H E$ and the $\\theta$-intermediate dimensions. The work clarifies the relationship between $\\mathcal{D}(E)$ and the lower dimension $\\dim_L E$, and discusses extensions and open questions. The results provide a robust measure-theoretic tool for multiscale fractal analysis with potential applications in projections and function images.
Abstract
We establish a unified Frostman-type framework connecting the classical Hausdorff dimension with the family of intermediate dimensions $\dim_θ$ recently introduced by Falconer, Fraser and Kempton. We define a new geometric quantity $\mathcal{D}(E)$ and prove that, under mild assumptions, there exists a family of measures $\{μ_δ\}$ supported on $E$ satisfying two simultaneous decay conditions, corresponding to the Hausdorff and intermediate Frostman inequalities. Such $(δ, s, t)$-Frostman measures allow for a two-scale characterization of the dimension of $E$.