[Paper Review] Stratified $β$-numbers and traveling salesman in Carnot groups

Sean Li|arXiv (Cornell University)|Feb 8, 2019

Mathematical Dynamics and Fractals14 references5 citations

TL;DR

This paper introduces stratified $\beta$-numbers as a refined metric tool for Carnot groups, enabling a complete characterization of 1-rectifiable sets via a traveling salesman theorem. By leveraging the group's stratified structure and developing new estimates on norm behavior and drift between lines, the authors prove that finite length rectifiable curves exist in any Carnot group if and only if the stratified $\beta$-energy and diameter are finite, resolving a key gap in prior work.

ABSTRACT

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

Motivation & Objective

• To resolve the lack of a complete traveling salesman theorem in Carnot groups by replacing standard $\beta$-numbers with a new, structure-sensitive variant.
• To establish a necessary and sufficient condition for a set in a Carnot group to lie on a finite-length rectifiable curve.
• To develop geometric and analytic tools tailored to the non-Euclidean, stratified structure of Carnot groups, particularly focusing on horizontal lines and homogeneous norms.
• To close the gap between necessary and sufficient conditions in previous results, especially in the Heisenberg group and higher-step Carnot groups.

Proposed method

• Introduce stratified $\beta$-numbers that measure the distance of a set to horizontal lines in each layer of the Carnot group's Lie algebra decomposition.
• Use the Hebisch-Sikora norm to define a homogeneous metric on the group, enabling precise control over distances and dilations.
• Develop new estimates on the 'drift' between almost-parallel horizontal line segments, exploiting the stratified structure of the $\beta$-numbers.
• Apply a Taylor expansion technique to the homogeneous norm to analyze curvature-like effects in the metric structure.
• Construct a sequence of abstract graphs approximating the set, then replace edges with quasi-convex curves to build a limiting rectifiable curve.
• Use a diagonalization argument on compact approximations to extract a limiting curve of finite length.

Experimental results

Research questions

• RQ1 Can a complete traveling salesman theorem be established in general Carnot groups using a modified notion of $\beta$-numbers?
• RQ2 What is the correct exponent for $\beta$-numbers in the integral condition to characterize 1-rectifiable sets in Carnot groups?
• RQ3 Why do previous $\beta$-number definitions fail to yield a sufficient condition in some Carnot groups, particularly the Heisenberg group?
• RQ4 How can the non-Euclidean geometry of Carnot groups be leveraged to control the behavior of distances and projections in the $\beta$-number estimates?

Key findings

• The authors introduce stratified $\beta$-numbers that are invariant under the group's dilations and capture the geometry of each layer in the Lie algebra decomposition.
• For any Carnot group, a set lies on a finite-length rectifiable curve if and only if the sum of its diameter and the integral of the $s$-th power of the stratified $\beta$-numbers is finite, where $s$ is the step of the group.
• In the Heisenberg group, the sufficient condition requires $p=4$, matching the necessary condition, thus closing the gap that existed in earlier work.
• The proof establishes that the integral of $\beta_{\text{strat}}^2$ over balls, weighted by $r^{-Q}dr$, is equivalent to the Carleson-type integral used in the Euclidean case.
• New estimates on the drift between almost-parallel horizontal lines are derived using the stratified $\beta$-numbers, which are essential for controlling the length of approximating curves.
• A Taylor expansion of the homogeneous norm allows precise control over the error in approximating distances in the group, which is critical for the length estimates.