The restricted content and the d-dimensional Analyst's Travelling Salesman Theorem for general sets

Abstract

In his 1990 paper, Jones characterized subsets of rectifiable curves in R2 via a multiscale sum of β-numbers, which measure how far a given set deviates from a straight line at each scale and location. This characterization was extended by Okikiolu to subsets of Rn and by Schul to subsets of a Hilbert space.Recently, there has been some interest in characterizing subsets of higher dimensional surfaces in Rn. Using a variant of Jones' β-number introduced by Azzam and Schul, Villa gave a characterization of lower regular subsets of a certain class of topologically stable surfaces – introduced in a 2004 paper of David – via a multiscale sum of these new β-numbers.In this paper we remove the lower regularity condition and prove an analogous result for general d-dimensional subsets of Rn. To do this, we introduce the restricted content, which assigns ‘mass’ to any subset of Rn (even to sets with zero Hausdorff measure), and use it to define new d-dimensional variant of Jones' β-number that is defined for any set in Rn.