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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.

Uniformly distributed measures have big pieces of Lipschitz graphs locally