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 \[ \mathscr{H}^1(\Gamma) \sim \text{diam}\, E + \sum_{Q} \beta_{E}(3Q)^2\ell(Q).\] 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} \beta_{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 $\mathbb{R}^n.$