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Quantitative metric density and connectivity for sets of positive measure

Volume 280 / 2025

Guy C. David, Brandon Oliva Studia Mathematica 280 (2025), 175-192 MSC: Primary 30L99; Secondary 28A75 DOI: 10.4064/sm240426-24-9 Published online: 5 December 2024

Abstract

We show that in doubling, geodesic metric measure spaces (including, for example, Euclidean space) sets of positive measure have a certain large-scale metric density property. As an application, we prove that a set of positive measure in the unit cube of $\mathbb R^d$ can be decomposed into a controlled number of subsets that are “well-connected” within the original set, along with a “garbage set” of arbitrarily small measure. Our results are quantitative, i.e., they provide bounds independent of the particular set under consideration.

Authors

  • Guy C. DavidDepartment of Mathematical Sciences
    Ball State University
    Muncie, IN 47306, USA
    e-mail
  • Brandon OlivaDepartment of Mathematics, Applied Mathematics, and Statistics
    Case Western Reserve University
    Cleveland, OH 44106-7058, USA
    e-mail

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