Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 44, 2019, 389-405

UNIFORMLY DISTRIBUTED MEASURES HAVE BIG PIECES OF LIPSCHITZ GRAPHS LOCALLY

A. Dali Nimer

University of Chicago, Department of Mathematics
5734, S. University Ave., Chicago, IL 60637, U.S.A.; nimer 'at' uchicago.edu

Abstract. The study of uniformly distributed measures was crucial in Preiss' proof of his theorem on rectifiability of measures with positive density. It is known that the support of a uniformly distributed measure is an analytic variety. In this paper, we provide quantitative information on the rectifiability of this variety. Tolsa had already shown that n-uniform measures have big pieces of Lipschitz graph (BPLG). Here, we prove that a uniformly distributed measure has BPLG locally.

2010 Mathematics Subject Classification: Primary 28A33, 49Q15.

Key words: Uniformly distributed measures, uniform rectifiability, big pieces of Lipschitz graphs.

Reference to this article: A. D. Nimer: Uniformly distributed measures have big pieces of Lipschitz graphs locally. Ann. Acad. Sci. Fenn. Math. 44 (2019), 389-405.

Full document as PDF file

https://doi.org/10.5186/aasfm.2019.4422