## 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

A related investigation on the Hausdorff dimension of projections of

*s*-sets onto lower-dimensional subspaces was carried out earlier by Marstrand [27].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.For the definition of and background on quasisymmetric maps, we refer the reader to [21].

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

## References

Alberti, G., Ottolini, M.: On the structure of continua with finite length and Golab’s semicontinuity theorem. Nonlinear Anal.

**153**, 35–55 (2017)Azzam, J., Schul, R.: An analyst’s traveling salesman theorem for sets of dimension larger than one. Math. Ann.

**370**(3–4), 1389–1476 (2018)Assouad, P.: Plongements lipschitziens dans \({ R}^{n}\). Bull. Soc. Math. France

**111**(4), 429–448 (1983)Azzam, J., Tolsa, X.: Characterization of \(n\)-rectifiability in terms of Jones’ square function: Part II. Geom. Funct. Anal.

**25**(5), 1371–1412 (2015)Besicovitch, A.S.: On the fundamental geometrical properties of linearly measurable plane sets of points. Math. Ann.

**98**(1), 422–464 (1928)Besicovitch, A.S.: On the fundamental geometrical properties of linearly measurable plane sets of points (II). Math. Ann.

**115**(1), 296–329 (1938)Bonk, M., Heinonen, J.: Smooth quasiregular mappings with branching. Publ. Math. Inst. Hautes Études Sci.

**100**(1), 153–170 (2004)Badger, M., Schul, R.: Multiscale analysis of 1-rectifiable measures: necessary conditions. Math. Ann.

**361**(3–4), 1055–1072 (2015)Badger, M., Schul, R.: Two sufficient conditions for rectifiable measures. Proc. Am. Math. Soc.

**144**(6), 2445–2454 (2016)Badger, M., Schul, R.: Multiscale analysis of 1-rectifiable measures II: characterizations. Anal. Geom. Metr. Spaces

**5**, 1–39 (2017)David, G., Semmes, S.: Singular integrals and rectifiable sets in \({ R}^n\): beyond Lipschitz graphs, Astérisque, no. 193 (1991)

David, G., Semmes, S.: Analysis Of and On Uniformly Rectifiable Sets. Mathematical Surveys and Monographs, vol. 38. American Mathematical Society, Providence (1993)

David, G., Semmes, S.: Fractured Fractals and Broken Dreams: Self-Similar Geometry Through Metric and Measure. Oxford Lecture Series in Mathematics and Its Applications, vol. 7. The Clarendon Press, Oxford University Press, New York (1997)

David, G., Toro, T.: Reifenberg flat metric spaces, snowballs, and embeddings. Math. Ann.

**315**(4), 641–710 (1999)David, G., Toro, T.: Reifenberg parameterizations for sets with holes. Mem. Am. Math. Soc.

**215**(1012), 6+102 (2012)Edelen, N., Naber, A., Valtorta, D.: Quantitative Reifenberg theorem for measures (2016). arXiv:1612.08052

Falconer, K.J.: The Geometry of Fractal Sets. Cambridge Tracts in Mathematics, vol. 85. Cambridge University Press, Cambridge (1986)

Federer, H.: The \((\varphi, k)\) rectifiable subsets of \(n\)-space. Trans. Am. Soc.

**62**, 114–192 (1947)Ghinassi, S.: A sufficient condition for \(C^{1,\alpha }\) parametrization (2017). arXiv:1709.06015

Garnett, J., Killip, R., Schul, R.: A doubling measure on \(\mathbb{R} ^d\) can charge a rectifiable curve. Proc. Am. Math. Soc.

**138**(5), 1673–1679 (2010)Heinonen, J.: Lectures on Analysis on Metric Spaces. Springer, New York (2001)

Hutchinson, J.E.: Fractals and self-similarity. Indiana Univ. Math. J.

**30**(5), 713–747 (1981)Jones, P.W.: Rectifiable sets and the traveling salesman problem. Invent. Math.

**102**(1), 1–15 (1990)Kechris, A.: Classical Descriptive Set Theory. Graduate Texts in Mathematics, vol. 156. Springer, New York (1995)

Luukkainen, J.: Assouad dimension: antifractal metrization, porous sets, and homogeneous measures. J. Korean Math. Soc.

**35**(1), 23–76 (1998)MacManus, P.: Catching sets with quasicircles. Rev. Mat. Iberoamericana

**15**(2), 267–277 (1999)Marstrand, J.M.: Some fundamental geometrical properties of plane sets of fractional dimensions. Proc. Lond. Math. Soc.

**4**, 257–302 (1954)Marstrand, J.M.: Hausdorff two-dimensional measure in \(3\)-space. Proc. Lond. Math. Soc.

**11**, 91–108 (1961)Mattila, P.: Hausdorff \(m\) regular and rectifiable sets in \(n\)-space. Trans. Am. Math. Soc.

**205**, 263–274 (1975)Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability. Cambridge Studies in Advanced Mathematics, vol. 44. Cambridge University Press, Cambridge (1995)

Martín, M.Á., Mattila, p: \(k\)-dimensional regularity classifications for \(s\)-fractals. Trans. Am. Math. Soc.

**305**(1), 293–315 (1988)Martín, M.A., Mattila, P.: Hausdorff measures, Hölder continuous maps and self-similar fractals. Math. Proc. Cambridge Philos. Soc.

**114**(1), 37–42 (1993)Martín, M.A., Mattila, P.: On the parametrization of self-similar and other fractal sets. Proc. Am. Math. Soc.

**128**(9), 2641–2648 (2000)Morse, A.P., Randolph, J.F.: The \(\phi \) rectifiable subsets of the plane. Trans. Am. Math. Soc.

**55**, 236–305 (1944)Mackay, J.M., Tyson, J.T.: Conformal Dimension: Theory and Application. University Lecture Series, vol. 54. American Mathematical Society, Providence (2010)

Okikiolu, K.: Characterization of subsets of rectifiable curves in \({ R}^n\). J. Lond. Math. Soc.

**46**(2), 336–348 (1992)Preiss, D.: Geometry of measures in \({ R}^n\): distribution, rectifiability, and densities. Ann. Math.

**125**(3), 537–643 (1987)Rushing, T.B.: Topological Embeddings. Pure and Applied Mathematics, vol. 52. Academic Press, New York (1973)

Romney, M., Vellis, V.: Bi-Lipschitz embedding of the generalized Grushin plane into Euclidean spaces. Math. Res. Lett. (2017)

Semmes, S.: Where the buffalo roam: infinite processes and infinite complexity (2003). arXiv:math/0302308 [math.CA]

Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, vol. 30. Princeton University Press, Princeton (1970)

Stein, E.M., Shakarchi, R.: Real Analysis, Measure Theory, Integration, and Hilbert Spaces. Princeton Lectures in Analysis, vol. 3. Princeton University Press, Princeton (2005)

Stong, R.: Mapping \(\mathbf{Z}^r\) into \(\mathbf{Z}^s\) with maximal contraction. Discrete Comput. Geom.

**20**(1), 131–138 (1998)Tolsa, X., Toro, T.: Rectifiability via a square function and Preiss’ theorem. Int. Math. Res. Not. (13), 4638–4662 (2015)

Väisälä, J.: Quasisymmetric embeddings in Euclidean spaces. Trans. Am. Math. Soc.

**264**(1), 191–204 (1981)Vellis, V.: Extension properties of planar uniform domains (2016). arXiv:1609.08763

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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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DOI: https://doi.org/10.1007/s12220-018-0034-2

### Keywords

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