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Given a metric space $X$, an Analyst's Traveling Salesman Theorem for $X$ gives a quantitative relationship between the length of a shortest curve containing any subset $E\subseteq X$ and a multi-scale sum measuring the ``flatness'' of $E$. The first such theorem was proven by Jones for $X = \mathbb{R}^2$ and extended to $X = \mathbb{R}^n$ by Okikiolu, while an analogous theorem was proven for Hilbert space, $X = H$, by Schul. Bishop has since shown that if one considers Jordan arcs, then the quantitative relationship given by Jones' and Okikioulu's results can be sharpened. This paper gives a full proof of Schul's original necessary half of the traveling salesman theorem in Hilbert space and provides a sharpening of the theorem's quantitative relationship when restricted to Jordan arcs analogous to Bishop's aforementioned sharpening in $\mathbb{R}^n$., Comment: 59 pages

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 $\mathbb{R}^n.$, Comment: 53 pages, 9 figures. Typos fixed and definition of "good cover" altered

We show that an Ahlfors $d$-regular set $E$ in $\mathbb{R}^{n}$ is uniformly rectifiable if the set of pairs $(x,r)\in E\times (0,\infty)$ for which there exists $y \in B(x,r)$ and $00$. To prove this, we generalize a result of Schul by proving, if $X$ is a $C$-doubling metric space, $\varepsilon,\rho\in (0,1)$, $A>1$, and $X_{n}$ is a sequence of maximal $2^{-n}$-separated sets in $X$, and $\mathscr{B}=\{B(x,2^{-n}):x\in X_{n},n\in \mathbb{N}\}$, then \[ \sum \left\{r_{B}^{s}: B\in \mathscr{B}, \frac{\mathscr{H}^{s}_{\rho r_{B}}(X\cap AB)}{(2r_{B})^{s}}>1+\varepsilon\right\} \lesssim_{C,A,\varepsilon,\rho,s} \mathscr{H}^{s}(X). \] This is a quantitative version of the classical result that for a metric space $X$ of finite $s$-dimensional Hausdorff measure, the upper $s$-dimensional densities are at most $1$ $\mathscr{H}^{s}$-almost everywhere., Comment: Fixed typos and minor errors, expanded the introduction, and corrected the counterexample on page 5

We generalize some characterizations of uniformly rectifiable (UR) sets to sets whose Hausdorff content is lower regular (and in particular, do not need to be Ahlfors regular). For example, David and Semmes showed that, given an Ahlfors $d$-regular set $E$, if we consider the set $\mathscr{B}$ of surface cubes (in the sense of Christ and David) near which $E$ does not look approximately like a union of planes, then $E$ is UR if and only if $\mathscr{B}$ satisfies a Carleson packing condition, that is, for any surface cube $R$, \[ \sum_{Q\subseteq R\atop Q\in \mathscr{B}} ({\rm diam} Q)^{d} \lesssim ({\rm diam} R)^{d}.\] We show that, for lower content regular sets that aren't necessarily Ahlfors regular, if $\beta_{E}(R)$ denotes the square sum of $\beta$-numbers over subcubes of $R$ as in the Traveling Salesman Theorem for higher dimensional sets [AS18], then \[ \mathscr{H}^{d}(R)+\sum_{Q\subseteq R\atop Q\in \mathscr{B}} ({\rm diam} Q)^{d}\sim \beta_{E}(R). \] We prove similar results for other uniform rectifiability critera, such as the Local Symmetry, Local Convexity, and Generalized Weak Exterior Convexity conditions. En route, we show how to construct a corona decomposition of any lower content regular set by Ahlfors regular sets, similar to the classical corona decomposition of UR sets by Lipschitz graphs developed by David and Semmes., Comment: 39 pages. To appear in Analysis & PDEs

In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable curves in the plane via a multiscale sum of $\beta$-numbers. These $\beta$-numbers are geometric quantities measuring how far a given set deviates from a best fitting line at each scale and location. Jones' result is a quantitative way of saying that a curve is rectifiable if and only if it has a tangent at almost every point. Moreover, computing this square sum for a curve returns the length of the curve up to multiplicative constant. K. Okikiolu extended his result from subsets of the plane to subsets of Euclidean space. G. David and S. Semmes extended the discussion to include sets of (integer) dimension larger than one, under the assumption of Ahlfors regularity and using a variant of Jones' $\beta$ numbers. In this paper we give a version of P. Jones' theorem for sets of arbitrary (integer) dimension lying in Euclidean space. We estimate the $d$-dimensional Hausdorff measure of a set in terms of an analogous sum of $\beta$-type numbers. There is no assumption of Ahlfors regularity, but rather, only of a lower bound on the Hausdorff content. We adapt David and Semmes' version of Jones' $\beta$-numbers by redefining them using a Choquet integral. A key tool in the proof is G. David and T. Toro's parametrization of Reifenberg flat sets (with holes)., Comment: Corrected more typos. There are still several typos and small mistakes in the published version of the paper, so the authors will maintain an up-to-date version on their webpages as we continue to correct them