Two-Scale Frostman Measures

TL;DR

This paper introduces a unified Frostman framework linking Hausdorff and intermediate dimensions through a new geometric quantity and two-scale measures, providing a novel characterization of fractal set dimensions.

Contribution

It develops a new geometric quantity and measures that connect classical and intermediate dimensions, offering a two-scale Frostman characterization of set dimensions.

Findings

Established a connection between Hausdorff and intermediate dimensions.

Defined a new geometric quantity $\

,

Abstract

We establish a unified Frostman-type framework connecting the classical Hausdorff dimension with the family of intermediate dimensions $\dim_\theta$ 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 $\{\mu_\delta\}$ supported on $E$ satisfying two simultaneous decay conditions, corresponding to the Hausdorff and intermediate Frostman inequalities. Such $(\delta, s, t)$-Frostman measures allow for a two-scale characterization of the dimension of $E$.