Stratified β$\beta$‐numbers and traveling salesman in Carnot groups

Abstract

We introduce a modified version of Jones's β $\beta$ -numbers for Carnot groups which we call stratified β $\beta$ -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 β $\beta$ -numbers in Carnot groups with stratified β $\beta$ -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 functions β $\beta$ 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 β $\beta$ -numbers must exhibit a gap between the necessary and sufficient directions.