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Geometry of Measures in Real Dimensions via Hölder Parameterizations

  • Matthew Badger
  • Vyron Vellis
Article

Abstract

We investigate the influence that s-dimensional lower and upper Hausdorff densities have on the geometry of a Radon measure in \(\mathbb {R}^n\) when s is a real number between 0 and n. This topic in geometric measure theory has been extensively studied when s is an integer. In this paper, we focus on the non-integer case, building upon a series of papers on s-sets by Martín and Mattila from 1988 to 2000. When \(0<s<1\), we prove that measures with almost everywhere positive lower density and finite upper density are carried by countably many bi-Lipschitz curves. When \(1\le s<n\), we identify conditions on the lower density that ensure the measure is either carried by or singular to (1 / s)-Hölder curves. The latter results extend part of the recent work of Badger and Schul, which examined the case \(s=1\) (Lipschitz curves) in depth. Of further interest, we introduce Hölder and bi-Lipschitz parameterization theorems for Euclidean sets with “small” Assouad dimension.

Keywords

Hölder parameterization Assouad dimension Uniformly disconnected sets Geometry of measures Hausdorff densities Generalized rectifiability 

Mathematics Subject Classification

Primary 28A75 Secondary 26A16, 30L05 

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Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ConnecticutStorrsUSA

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