Intermediate dimensions of measures: Interpolating between Hausdorff and Minkowski dimensions

TL;DR

This paper introduces a new family of intermediate dimensions for Borel measures that bridge the gap between Hausdorff and Minkowski dimensions, expanding the understanding of measure dimensions.

Contribution

It defines a novel set of intermediate dimensions for measures, filling a gap between existing dimensions like Fourier, Sobolev, and Assouad, and provides capacity-based estimation methods.

Findings

Established a new family of measure dimensions between Hausdorff and Minkowski.

Provided a capacity-theoretic framework for estimating dimensions of pushforward measures.

Connected intermediate dimensions with a reverse Frostman lemma.

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.