  A natural quantity that measures how well a map $f:\mathbb{R}^{d}\rightarrow
\mathbb{R}^{D}$ is approximated by an affine transformation is
\[\omega_{f}(x,r)=\inf_{A}\left(\frac{1}{|B(x,r)|}\int_{B(x,r)}\left(\frac{|f-A|}{|A'|r}\right)^{2}\right)^{\frac{1}{2}},\]
where the infimum ranges over all non constant affine transformations. This is
natural insofar as it is invariant under rescaling $f$ in either its domain or
image. We show that if $f:\mathbb{R}^{d}\rightarrow \mathbb{R}^{D}$ is
quasisymmetric and its image has a sufficient amount of rectifiable structure
(although not necessarily $\mathcal{H}^{d}$-finite), then
$\omega_{f}(x,r)^{2}\frac{dxdr}{r}$ is a Carleson measure on
$\mathbb{R}^{d}\times(0,\infty)$. Moreover, this is an equivalence: the
existence of such a Carleson measure implies that, in every ball
$B(x,r)\subseteq \mathbb{R}^{d}$, there is a set $E$ occupying 90$%$ of
$B(x,r)$, say, upon which $f$ is bi-Lipschitz (and hence guaranteeing
rectifiable pieces in the image).
  En route, we make a minor adjustment to a theorem of Semmes to show that
quasisymmetric maps of subsets of $\mathbb{R}^{d}$ into $\mathbb{R}^{d}$ are
bi-Lipschitz on a large subset quantitatively.
