# Geometry of Measures in Real Dimensions via Hölder Parameterizations

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## 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## References

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