Quantitative comparisons of multiscale geometric properties

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