What is the definition of uniform rectifiability as used in the context of analytic capacity of compact sets in $\mathbb{C}$?
The precise context is this paper by Mattila, Mernikov and Verdera.
2026-03-28 12:38:30.1774701510
Uniform Rectifiability
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A closed set $E\subset\mathbb{C}$ is called uniformly rectifiable if there exists a curve $E\subset\Gamma\subset\mathbb{C}$ and $C>0$ such that $$\mathcal{H}^1(\Gamma\cap B(z,r))\le Cr$$ for $z\in\Gamma$ and $r>0$ where $\mathcal{H}^1$ denotes 1-dimensional Hausdorff measure and $B(z,r)=\{w\in\mathbb{C}\,:\,|z-w|\le r\}$.