2025/02/02 by Nicolas Angelini, Angelini, Nicolas E., Ursula Molter +2 · 1 citation
Computer Science · Mathematics · #Classical Analysis and ODEs (math.CA) #Digital Image Processing Techniques #FOS: Mathematics #Mathematical Dynamics and Fractals #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2502.00772
openalex publication_date 2025/02/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31
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.