Intermediate dimensions of measures: Interpolating between Hausdorff and Minkowski 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 [10] defined intermediate dimensions that interpolate between the Minkowski and Assouad dimensions for measures. Additionally, Fraser, in [7] introduced intermediate dimensions that interpolate between the Fourier and Hausdorff 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 3.11 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.