A $d$-dimensional Analyst's Travelling Salesman Theorem for general sets in ${\mathbb{R}}^n$

Abstract

In his 1990 paper, Jones proved the following: given $E\subseteq {\mathbb{R}}^2$, there exists a curve $\Gamma$ such that $E\subseteq \Gamma$ and \[ \mathscr{H}^1(Γ) \sim \text{diam}\, E + \sum_{Q} β_{E}(3Q)^2\ell(Q).\] Here, ${\beta }_E(Q)$ measures how far $E$ deviates from a straight line inside $Q$. This was extended by Okikiolu to subsets of ${\mathbb{R}}^n$ and by Schul to subsets of a Hilbert space. In 2018, Azzam and Schul introduced a variant of the Jones $\beta$-number. With this, they, and separately Villa, proved similar results for lower regular subsets of ${\mathbb{R}}^n.$ In particular, Villa proved that, given $E\subseteq {\mathbb{R}}^n$ which is lower content regular, there exists a `nice' $d$-dimensional surface $F$ such that $E\subseteq F$ and \begin{align} \mathscr{H}^d(F) \sim \text{diam}( E)^d + \sum_{Q} β_{E}(3Q)^2\ell(Q)^d. \end{align} In this context, a set $F$ is `nice' if it satisfies a certain topological non degeneracy condition, first introduced in a 2004 paper of David. In this paper we drop the lower regularity condition and prove an analogous result for general $d$-dimensional subsets of ${\mathbb{R}}^n.$ To do this, we introduce a new $d$-dimensional variant of the Jones $\beta$-number that is defined for any set in ${\mathbb{R}}^n.$

53 pages, 9 figures. Typos fixed and definition of "good cover" altered