  We generalize some characterizations of uniformly rectifiable (UR) sets to
sets whose Hausdorff content is lower regular (and in particular, do not need
to be Ahlfors regular). For example, David and Semmes showed that, given an
Ahlfors $d$-regular set $E$, if we consider the set $\mathscr{B}$ of surface
cubes (in the sense of Christ and David) near which $E$ does not look
approximately like a union of planes, then $E$ is UR if and only if
$\mathscr{B}$ satisfies a Carleson packing condition, that is, for any surface
cube $R$, \[ \sum_{Q\subseteq R\atop Q\in \mathscr{B}} ({\rm diam} Q)^{d}
\lesssim ({\rm diam} R)^{d}.\] We show that, for lower content regular sets
that aren't necessarily Ahlfors regular, if $\beta_{E}(R)$ denotes the square
sum of $\beta$-numbers over subcubes of $R$ as in the Traveling Salesman
Theorem for higher dimensional sets [AS18], then \[
\mathscr{H}^{d}(R)+\sum_{Q\subseteq R\atop Q\in \mathscr{B}} ({\rm diam}
Q)^{d}\sim \beta_{E}(R). \] We prove similar results for other uniform
rectifiability critera, such as the Local Symmetry, Local Convexity, and
Generalized Weak Exterior Convexity conditions.
  En route, we show how to construct a corona decomposition of any lower
content regular set by Ahlfors regular sets, similar to the classical corona
decomposition of UR sets by Lipschitz graphs developed by David and Semmes.
