## 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.

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## Notes

- 1.
A related investigation on the Hausdorff dimension of projections of

*s*-sets onto lower-dimensional subspaces was carried out earlier by Marstrand [27]. - 2.
In fact,

*K*is Ahlfors regular in the sense that \(\mathcal {H}^m(K\cap B(x,r))\sim r^m\) when \(x\in K\) and \(0<r\le \mathrm{diam}\,K\), because*K*is self-similar. - 3.
For the definition of and background on quasisymmetric maps, we refer the reader to [21].

- 4.
Constructions of tree-like surfaces are by now classical. For instance, see [38, Figure 2.4.16].

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## Additional information

Badger was partially supported by NSF Grants 1500382 and 1650546. Part of this work was carried out while Badger attended the long program on *Harmonic Analysis* at MSRI in Spring 2017.

## Appendix A: Decomposition of \(\sigma \)-Finite Measures

### Appendix A: Decomposition of \(\sigma \)-Finite Measures

The following definition encodes commonly used definitions of countably rectifiable and purely unrectifiable measures, including the variants in Definition 1.1.

### Definition A.1

Let \((\mathbb {X},\mathcal {M})\) be a measurable space, let \(\mathcal {N}\subseteq \mathcal {M}\) be a non-empty collection of measurable sets, and let \(\mu \) be a measure defined on \((\mathbb {X},\mathcal {M})\). We say that \(\mu \) is *carried by*\(\mathcal {N}\) provided there exists a countable family \(\{\Gamma _i:i\ge 1\}\subseteq \mathcal {N}\) of sets with

We say that \(\mu \) is *singular to*\(\mathcal {N}\) if \(\mu (\Gamma )=0\) for every \(\Gamma \in \mathcal {N}\).

The “correctness” of Definition A.1 is partially justified by the following proposition, which should be considered a standard exercise in measure theory. The proof is a slight variation of [10, Proposition 1.1] (or [30, Theorem 15.6]), which is specialized to the decomposition of Radon measures (sets) in \(\mathbb {R}^n\) into countably *m*-rectifiable and purely *m*-unrectifiable components. We present details for the convenience of the reader.

### Proposition A.2

(Decomposition) Let \((\mathbb {X},\mathcal {M})\) be a measurable space and let \(\mathcal {N}\subseteq \mathcal {M}\) be a non-empty collection of sets. If \(\mu \) is a \(\sigma \)-finite measure on \((\mathbb {X},\mathcal {M})\), then \(\mu \) can be written uniquely as

where \(\mu _\mathcal {N}\) is a measure on \((\mathbb {X},\mathcal {M})\) that is carried by \(\mathcal {N}\) and \(\mu ^\perp _\mathcal {N}\) is a measure on \((\mathbb {X},\mathcal {M})\) that is singular to \(\mathcal {N}\).

### Proof

Let \(\widetilde{\mathcal {N}}\) denote the collection of finite unions of sets in \(\mathcal {N}\). Given a \(\sigma \)-finite measure \(\mu \) on \((\mathbb {X},\mathcal {M})\), expand \(\mathbb {X}=\bigcup _{j=1}^\infty X_j\), where

is an increasing chain of sets in \(\mathcal {M}\) with \(\mu (X_j)<\infty \) for all \(j\ge 1\). For each \(j\ge 1\), define

By the approximation property of the supremum, we may choose a sequence \((N_j)_{j=1}^\infty \) of sets in \(\widetilde{\mathcal {N}}\) such that \(\mu (X_j\cap N_j)>M_j-1/j\) for all \(j\ge 1\). Fix any such \((N_j)_{j=1}^\infty \) and define

Then \(\mu _\mathcal {N}\) and \(\mu ^\perp _\mathcal {N}\) are measures on \((\mathbb {X},\mathcal {M})\) with \(\mu =\mu _\mathcal {N}+\mu _\mathcal {N}^\perp \) and it is clear that \(\mu _{\mathcal {N}}\) is carried by \(\mathcal {N}\).

To see that \(\mu ^\perp _\mathcal {N}\) is singular to \(\mathcal {N}\), assume for contradiction that \(\mu ^\perp _{\mathcal {N}}(S)>0\) for some \(S\in \mathcal {N}\). First pick an index \(j_0\) such that \(\mu (X_{j_0}\cap S)>0\). Next, pick \(j\ge j_0\) sufficiently large such that \(\mu (X_{j_0}\cap S)>1/j\). Note that \(T:=N_j\cup S\in \widetilde{\mathcal {N}}\), since \(N_j\in \widetilde{\mathcal {N}}\) and \(S\in \mathcal {N}\). It follows that

where in the last inequality we used the fact that \(X_{j_0}\subseteq X_j\). We have a reached a contradiction. Therefore, \(\mu ^\perp _\mathcal {N}\) is singular to \(\mathcal {N}\).

Next we want to show that the decomposition of \(\mu \) as the sum of a measure that is carried by \(\mathcal {N}\) and a measure that is singular to \(\mathcal {N}\) is unique. Suppose that \(\mu =\mu _c+\mu _s\), where \(\mu _c\) and \(\mu _s\) are measures such that \(\mu _c\) is carried by \(\mathcal {N}\) and \(\mu _s\) is singular to \(\mathcal {N}\). To show that \(\mu _c=\mu _\mathcal {N}\) and \(\mu _s=\mu ^\perp _\mathcal {N}\), it suffices to prove the former. Suppose for contradiction that \(\mu _c(A)<\mu _\mathcal {N}(A)\) for some \(A\in \mathcal {M}\). Replacing *A* with \(A\cap X_j\) for *j* sufficiently large, we may assume without loss of generality that \(\mu _{\mathcal {N}}(A)<\infty \). Since \(\mu _c\) and \(\mu _\mathcal {N}\) are both carried by \(\mathcal {N}\), we can find a set *N*, which is a countable union of sets in \(\mathcal {N}\) such that

Then \(\mu _s(A\cap N)=\mu (A\cap N)-\mu _c(A\cap N) > \mu (A\cap N)-\mu _\mathcal {N}(A\cap N)=\mu ^\perp _\mathcal {N}(A\cap N)=0.\) This contradicts that \(\mu _s\) is singular to \(\mathcal {N}\). Therefore, \(\mu _c=\mu _\mathcal {N}\), and thus, \(\mu _s=\mu ^\perp _\mathcal {N}\).\(\square \)

### Example A.3

Let \(\mu \) and \(\nu \) be measures on a measurable space \((\mathbb {X},\mathcal {M})\), and let

denote the null sets of \(\nu \). If \(\mu \) is \(\sigma \)-finite, then by Proposition A.2, the measure \(\mu \) can be uniquely expanded \(\mu =\mu _\mathcal {N}+\mu _{\mathcal {N}}^\perp \), where \(\mu _{\mathcal {N}}\) is carried by null sets of \(\nu \) and \(\mu _{\mathcal {N}}^\perp \) is singular to null sets of \(\nu \). Thus, writing \(\mu _s:=\mu _{\mathcal {N}}\) and \(\mu _{ac}:=\mu _{\mathcal {N}}^\perp \), we can decompose \(\mu =\mu _{s}+\mu _{ac}\), where \(\mu _{s}\perp \nu \) and \(\mu _{ac}\ll \nu \).

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Badger, M., Vellis, V. Geometry of Measures in Real Dimensions via Hölder Parameterizations.
*J Geom Anal* **29, **1153–1192 (2019). https://doi.org/10.1007/s12220-018-0034-2

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### Keywords

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

### Mathematics Subject Classification

- Primary 28A75
- Secondary 26A16, 30L05