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arXiv · 2502.00772

Intermediate dimensions of measures: Interpolating between Hausdorff and Minkowski dimensions

Nicolas E. Angelini·Ursula M. Molter·Jose M. Tejada
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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.

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Nicolas E. Angelini, Ursula M. Molter, Jose M. Tejada. 2025-02-02. Intermediate dimensions of measures: Interpolating between Hausdorff and Minkowski dimensions. https://doi.org/10.1016/j.jmaa.2025.130039

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