Small constant uniform rectifiability

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.