Intermediate dimensions of measures: Interpolating between Hausdorff and Minkowski dimensions
Nicolas E. Angelini, Ursula M. Molter, Jose M. Tejada
TL;DR
This work introduces intermediate dimensions for finite Borel measures that interpolate between the upper Hausdorff dimension $\dim_H^*(\mu)$ and Minkowski dimension $\overline{\dim}_B(\mu)$, filling a gap in the spectrum of measure dimensions. It develops two parallel formulations: a direct ball-based definition via $\overline{\dim}_\theta(\mu)$ and $\underline{\dim}_\theta(\mu)$, and a capacity-theoretic framework using kernels $\phi_{r,\theta}^{s,m}$ that yield equivalent threshold characterizations. The paper establishes foundational properties, including endpoint identifications, monotonicity, product behavior, and projection invariance, and proves a reverse-Frostman-type lemma in this intermediate setting. It further analyzes limit behavior as $\theta\to0$, elucidating how projections of measures preserve or reflect the intermediate dimension structure, with implications for pushforward measures and capacity-based dimensions.
Abstract
In this paper, we define a family of dimensions for Borel measures that lie between the Hausdorff and Minkowski dimensions for measures, analogous to the intermediate dimensions of sets. Previously, Hare et. al. in [11] defined families of dimensions that interpolate between the Minkowski and Assouad dimensions for measures. Additionally, Fraser, in [8] introduced an additional family of dimensions that interpolate between the Fourier and Sobolev dimensions of measures. Our results address a "gap" in the study of dimension interpolation for measures, almost completing the spectrum of intermediate dimensions for measures: from Fourier to Assouad dimensions. Furthermore, Theorem 3.13 can be interpreted as a "reverse Frostman" lemma for intermediate dimensions. We also obtain a capacity-theoretic definition that enables us to estimate the intermediate dimensions of pushforward measures by projections.