# Geometry of Measures in Real Dimensions via Hölder Parameterizations

## 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

- 1.Alberti, G., Ottolini, M.: On the structure of continua with finite length and Golab’s semicontinuity theorem. Nonlinear Anal.
**153**, 35–55 (2017)MathSciNetCrossRefzbMATHGoogle Scholar - 2.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)MathSciNetCrossRefzbMATHGoogle Scholar - 3.Assouad, P.: Plongements lipschitziens dans \({ R}^{n}\). Bull. Soc. Math. France
**111**(4), 429–448 (1983)MathSciNetCrossRefzbMATHGoogle Scholar - 4.Azzam, J., Tolsa, X.: Characterization of \(n\)-rectifiability in terms of Jones’ square function: Part II. Geom. Funct. Anal.
**25**(5), 1371–1412 (2015)MathSciNetCrossRefzbMATHGoogle Scholar - 5.Besicovitch, A.S.: On the fundamental geometrical properties of linearly measurable plane sets of points. Math. Ann.
**98**(1), 422–464 (1928)MathSciNetCrossRefzbMATHGoogle Scholar - 6.Besicovitch, A.S.: On the fundamental geometrical properties of linearly measurable plane sets of points (II). Math. Ann.
**115**(1), 296–329 (1938)MathSciNetCrossRefzbMATHGoogle Scholar - 7.Bonk, M., Heinonen, J.: Smooth quasiregular mappings with branching. Publ. Math. Inst. Hautes Études Sci.
**100**(1), 153–170 (2004)MathSciNetCrossRefzbMATHGoogle Scholar - 8.Badger, M., Schul, R.: Multiscale analysis of 1-rectifiable measures: necessary conditions. Math. Ann.
**361**(3–4), 1055–1072 (2015)MathSciNetCrossRefzbMATHGoogle Scholar - 9.Badger, M., Schul, R.: Two sufficient conditions for rectifiable measures. Proc. Am. Math. Soc.
**144**(6), 2445–2454 (2016)MathSciNetCrossRefzbMATHGoogle Scholar - 10.Badger, M., Schul, R.: Multiscale analysis of 1-rectifiable measures II: characterizations. Anal. Geom. Metr. Spaces
**5**, 1–39 (2017)MathSciNetCrossRefzbMATHGoogle Scholar - 11.David, G., Semmes, S.: Singular integrals and rectifiable sets in \({ R}^n\): beyond Lipschitz graphs, Astérisque, no. 193 (1991)Google Scholar
- 12.David, G., Semmes, S.: Analysis Of and On Uniformly Rectifiable Sets. Mathematical Surveys and Monographs, vol. 38. American Mathematical Society, Providence (1993)zbMATHGoogle Scholar
- 13.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)zbMATHGoogle Scholar
- 14.David, G., Toro, T.: Reifenberg flat metric spaces, snowballs, and embeddings. Math. Ann.
**315**(4), 641–710 (1999)MathSciNetCrossRefzbMATHGoogle Scholar - 15.David, G., Toro, T.: Reifenberg parameterizations for sets with holes. Mem. Am. Math. Soc.
**215**(1012), 6+102 (2012)Google Scholar - 16.Edelen, N., Naber, A., Valtorta, D.: Quantitative Reifenberg theorem for measures (2016). arXiv:1612.08052
- 17.Falconer, K.J.: The Geometry of Fractal Sets. Cambridge Tracts in Mathematics, vol. 85. Cambridge University Press, Cambridge (1986)zbMATHGoogle Scholar
- 18.Federer, H.: The \((\varphi, k)\) rectifiable subsets of \(n\)-space. Trans. Am. Soc.
**62**, 114–192 (1947)MathSciNetzbMATHGoogle Scholar - 19.Ghinassi, S.: A sufficient condition for \(C^{1,\alpha }\) parametrization (2017). arXiv:1709.06015
- 20.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)CrossRefzbMATHGoogle Scholar - 21.Heinonen, J.: Lectures on Analysis on Metric Spaces. Springer, New York (2001)CrossRefzbMATHGoogle Scholar
- 22.Hutchinson, J.E.: Fractals and self-similarity. Indiana Univ. Math. J.
**30**(5), 713–747 (1981)MathSciNetCrossRefzbMATHGoogle Scholar - 23.Jones, P.W.: Rectifiable sets and the traveling salesman problem. Invent. Math.
**102**(1), 1–15 (1990)MathSciNetCrossRefzbMATHGoogle Scholar - 24.Kechris, A.: Classical Descriptive Set Theory. Graduate Texts in Mathematics, vol. 156. Springer, New York (1995)zbMATHGoogle Scholar
- 25.Luukkainen, J.: Assouad dimension: antifractal metrization, porous sets, and homogeneous measures. J. Korean Math. Soc.
**35**(1), 23–76 (1998)MathSciNetzbMATHGoogle Scholar - 26.MacManus, P.: Catching sets with quasicircles. Rev. Mat. Iberoamericana
**15**(2), 267–277 (1999)MathSciNetCrossRefzbMATHGoogle Scholar - 27.Marstrand, J.M.: Some fundamental geometrical properties of plane sets of fractional dimensions. Proc. Lond. Math. Soc.
**4**, 257–302 (1954)MathSciNetCrossRefzbMATHGoogle Scholar - 28.Marstrand, J.M.: Hausdorff two-dimensional measure in \(3\)-space. Proc. Lond. Math. Soc.
**11**, 91–108 (1961)MathSciNetCrossRefzbMATHGoogle Scholar - 29.Mattila, P.: Hausdorff \(m\) regular and rectifiable sets in \(n\)-space. Trans. Am. Math. Soc.
**205**, 263–274 (1975)MathSciNetzbMATHGoogle Scholar - 30.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)CrossRefzbMATHGoogle Scholar
- 31.Martín, M.Á., Mattila, p: \(k\)-dimensional regularity classifications for \(s\)-fractals. Trans. Am. Math. Soc.
**305**(1), 293–315 (1988)MathSciNetCrossRefzbMATHGoogle Scholar - 32.Martín, M.A., Mattila, P.: Hausdorff measures, Hölder continuous maps and self-similar fractals. Math. Proc. Cambridge Philos. Soc.
**114**(1), 37–42 (1993)MathSciNetCrossRefzbMATHGoogle Scholar - 33.Martín, M.A., Mattila, P.: On the parametrization of self-similar and other fractal sets. Proc. Am. Math. Soc.
**128**(9), 2641–2648 (2000)MathSciNetCrossRefzbMATHGoogle Scholar - 34.Morse, A.P., Randolph, J.F.: The \(\phi \) rectifiable subsets of the plane. Trans. Am. Math. Soc.
**55**, 236–305 (1944)MathSciNetCrossRefzbMATHGoogle Scholar - 35.Mackay, J.M., Tyson, J.T.: Conformal Dimension: Theory and Application. University Lecture Series, vol. 54. American Mathematical Society, Providence (2010)zbMATHGoogle Scholar
- 36.Okikiolu, K.: Characterization of subsets of rectifiable curves in \({ R}^n\). J. Lond. Math. Soc.
**46**(2), 336–348 (1992)MathSciNetCrossRefzbMATHGoogle Scholar - 37.Preiss, D.: Geometry of measures in \({ R}^n\): distribution, rectifiability, and densities. Ann. Math.
**125**(3), 537–643 (1987)MathSciNetCrossRefzbMATHGoogle Scholar - 38.Rushing, T.B.: Topological Embeddings. Pure and Applied Mathematics, vol. 52. Academic Press, New York (1973)zbMATHGoogle Scholar
- 39.Romney, M., Vellis, V.: Bi-Lipschitz embedding of the generalized Grushin plane into Euclidean spaces. Math. Res. Lett. (2017)Google Scholar
- 40.Semmes, S.: Where the buffalo roam: infinite processes and infinite complexity (2003). arXiv:math/0302308 [math.CA]
- 41.Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, vol. 30. Princeton University Press, Princeton (1970)zbMATHGoogle Scholar
- 42.Stein, E.M., Shakarchi, R.: Real Analysis, Measure Theory, Integration, and Hilbert Spaces. Princeton Lectures in Analysis, vol. 3. Princeton University Press, Princeton (2005)zbMATHGoogle Scholar
- 43.Stong, R.: Mapping \(\mathbf{Z}^r\) into \(\mathbf{Z}^s\) with maximal contraction. Discrete Comput. Geom.
**20**(1), 131–138 (1998)MathSciNetCrossRefGoogle Scholar - 44.Tolsa, X., Toro, T.: Rectifiability via a square function and Preiss’ theorem. Int. Math. Res. Not. (13), 4638–4662 (2015)Google Scholar
- 45.Väisälä, J.: Quasisymmetric embeddings in Euclidean spaces. Trans. Am. Math. Soc.
**264**(1), 191–204 (1981)MathSciNetCrossRefzbMATHGoogle Scholar - 46.Vellis, V.: Extension properties of planar uniform domains (2016). arXiv:1609.08763