Quantitative comparisons of multiscale geometric properties (Q2236625)

From the abstract: ``We generalize some characterizations of uniformly rectifiable sets whose Hausdorff content is lower regular (and, in particular, is not necessarily Ahlfors regular).'' Many of these characterizations originate in work of \textit{G. David} and \textit{S. Semmes} [Singular integrals and rectifiable sets in \(R^ n\). Au-delà des graphes lipschitziens. Montrouge: Société Mathématique de France (1991; Zbl 0743.49018); Analysis of and on uniformly rectifiable sets. Providence, RI: American Mathematical Society (1993; Zbl 0832.42008)] and involve estimates of Hausdorff measures of various dimensions.\par Unfortunately there seems to be no good way to review this paper without simultaneously boring experts and overwhelming outsiders with the definitions of the notions that are being studied and employed. Readers familiar with terms like `Ahlfors regularity', `uniform rectifiability', `Carleson packing condition', `the analyst's traveling salesman theorem, and others will find much here that will interest them, while those of a different bent would need to invest some time to get an appreciation for the results.