An Analyst's Traveling Salesman Theorem for sets of dimension larger than one

TL;DR

This paper extends Jones' Traveling Salesman Theorem to sets of any integer dimension in Euclidean space, providing a new way to estimate their Hausdorff measure using generalized beta numbers without requiring Ahlfors regularity.

Contribution

It introduces a novel version of Jones' theorem for higher-dimensional sets with minimal regularity assumptions, utilizing a Choquet integral approach to beta numbers.

Findings

Estimates the Hausdorff measure via beta-type sums for higher-dimensional sets.

Removes the Ahlfors regularity assumption, only requiring a lower Hausdorff content bound.

Employs Reifenberg flat set parametrization in the proof.

Abstract

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).