Small constant uniform rectifiability

TL;DR

This paper characterizes uniformly rectifiable sets in Euclidean space with densities near one, establishing new equivalences and estimates that deepen understanding of geometric measure theory and chord-arc domains.

Contribution

It provides new equivalent characterizations of uniformly rectifiable sets with densities close to one, including a novel Carleson measure estimate for Tolsa alpha coefficients.

Findings

Reifenberg flat sets with small constants have small Tolsa alpha coefficients.

New characterization of chord-arc domains with small constants.

Establishes equivalences for uniformly rectifiable sets with near-unit density.

Abstract

We provide several equivalent characterizations of locally flat, $d$-Ahlfors regular, uniformly rectifiable sets $E$ in $\mathbb{R}^n$ with density close to $1$ for any dimension $d \in \mathbb{N}$ with $1 \le d \le n-1$. In particular, we show that when $E$ is Reifenberg flat with small constant and has Ahlfors regularity constant close to $1$, then the Tolsa alpha coefficients associated to $E$ satisfy a small constant Carleson measure estimate. This estimate is new, even when $d = n-1$, and gives a new characterization of chord-arc domains with small constant.