Two-Scale Frostman Measures

2025/11/06 by Nicolas Angelini, Angelini, Nicolas, Ursula Molter +1
Mathematics · #Advanced Operator Algebra Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Geometry and complex manifolds #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2511.04302

openalex publication_date 2025/11/06 · openalex created_date 2025/11/08 · openalex updated_date 2026/07/28

Abstract

We establish a unified Frostman-type framework connecting the classical Hausdorff dimension with the family of intermediate dimensions dim_θ recently introduced by Falconer, Fraser and Kempton. We define a new geometric quantity D(E) and prove that, under mild assumptions, there exists a family of measures \μ_δ\ supported on E satisfying two simultaneous decay conditions, corresponding to the Hausdorff and intermediate Frostman inequalities. Such (δ, s, t)-Frostman measures allow for a two-scale characterization of the dimension of E.

Citations

• Intermediate dimensions of measures: Interpolating between Hausdorff and Minkowski dimensions
• Critical values for Intermediate and Box dimension of projections and other images
• Intermediate dimensions of complementary sets
• Fractal Geometry: Mathematical Foundations and Applications.

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