  We introduce new flatness coefficients, which we call \emph{$\iota$-numbers},
for Ahlfors regular sets in metric spaces. We investigate the relation between
Carleson-type geometric lemmas for $\iota$-numbers and other concepts related
to quantitative rectifiability. We characterize uniform $1$-rectifiability in
rather general metric spaces, and uniform $k$-rectfiability in Euclidean
spaces, using $\iota$-numbers. The characterization of uniform
$1$-rectifiability in the metric setting proceeds by quantifying an isometric
embedding theorem due to Menger, and by an abstract argument that allows to
pass from a local covering by continua to a global covering by $1$-regular
connected sets. The Euclidean result, on the other hand, is a consequence of an
abstract statement about approximation by generalized planes, which applies
also to the study of low-dimensional sets in Heisenberg groups.
