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 \(d \in {\mathbb {N}}\) , \(1 \le d < n\) . 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.
