Sets with topology, the Analyst's TST, and applications

Michele Villa

Published 2019-08-27Version 1

This paper was motivated by three questions. First: in a recent paper, Azzam and Schul asked what sort of sets could play the role of curves in the context of the higher dimensional analyst's traveling salesman theorem. Second: given a set in the euclidean space which has some `lower bound' on its topology, and some upper bound on its size (in terms of Hausdorff measure), what can we say about its geometric complexity? This was initially raised by Semmes in the mid-nineties. Third: in a paper from 1997, Bishop and Jones proved that if a connected set in the plane is uniformly non-flat (the non-flatness being quantified in terms of the Jones $\beta$ coefficients), then its dimension must be strictly larger than one | how much larger depending on how non-flat the set is; can one prove a similar result for higher dimensional sets? In this paper we try to give some answers to these questions. We show that if put on $E$ a certain topological non degeneracy condition, introduced by David in a paper from 2004, giving $E$ a robust $d$-dimensionality, then, first, $E$ will satisfy an analyst's traveling salesman type quantitative estimate, that is, an estimate that looks like $ \mathcal{H}^d(E) \sim \sum_{Q} \beta_E^2(Q) \ell(Q)^d + \text{diam}(E)^d. $ Second, if we also assume that $E$ is upper Ahlfors regular, then $E$ is uniformly rectifiable. Third, we prove an exact analogue of the theorem of Bishop and Jones, with an explicit dependency of the dimensional lower bound to the non-flatness parameter.