Uniformly Distributed Measures have Big Pieces of Lipschitz Graphs locally

TL;DR

This paper demonstrates that the support of locally uniformly distributed measures contains large regions that can be approximated by Lipschitz graphs, advancing understanding of measure rectifiability.

Contribution

It extends previous results by showing that uniformly distributed measures have Big Pieces of Lipschitz Graphs locally, providing quantitative rectifiability insights.

Findings

Uniformly distributed measures have locally Big Pieces of Lipschitz Graphs.

Supports of these measures are quantitatively rectifiable.

Advances understanding of measure structure in geometric measure theory.

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 Graphs(BPLG) . Here, we prove that a uniformly distributed measure has BPLG locally.