Stratified $\beta$-numbers and traveling salesman in Carnot groups

TL;DR

This paper extends Jones's traveling salesman theorem to Carnot groups using a new stratified version of $eta$-numbers, providing a characterization of rectifiable curves and addressing limitations of previous approaches.

Contribution

It introduces stratified $eta$-numbers for Carnot groups and proves an analogue of Jones's theorem, expanding the understanding of rectifiability in these groups.

Findings

Stratified $eta$-numbers effectively characterize rectifiable sets in Carnot groups.

The proof involves new estimates on the drift between almost parallel line segments.

An example shows limitations of unmodified $eta$-numbers in certain Carnot groups.

Abstract

We introduce a modified version of P. Jones's $\beta$-numbers for Carnot groups which we call {\it stratified $\beta$-numbers}. We show that an analogue of Jones's traveling salesman theorem on 1-rectifiability of sets holds for any Carnot group if we replace previous notions of $\beta$-numbers in Carnot groups with stratified $\beta$-numbers. As we generalize both directions of the traveling salesman theorem, we get a characterization of subsets of Carnot groups that lie on finite length rectifiable curves. Our proof expands upon previous analysis of the Hebisch-Sikora norm for Carnot groups. In particular, we find new estimates on the drift between almost parallel line segments that take advantage of the stratified $\beta$'s and also develop a Taylor expansion technique of the norm. We also give an example of a Carnot group for which a traveling salesman theorem based on the unmodified $\beta$-numbers must exhibit a gap between the necessary and sufficient directions.