An upper bound for the length of a Traveling Salesman path in the Heisenberg group

TL;DR

This paper establishes an improved upper bound for the length of a Traveling Salesman path in the Heisenberg group by refining geometric conditions involving Jones-$eta$-numbers, extending previous results.

Contribution

It introduces a new sufficient condition using a modified geometric lemma with a lower power of Jones-$eta$-numbers, improving previous bounds in the Heisenberg group context.

Findings

Improved upper bound on TSP path length in the Heisenberg group.

Replaced power 2 with any power less than 4 in Jones-$eta$-number estimates.

Extended the geometric conditions for rectifiable curves in sub-Riemannian geometry.

Abstract

We show that a sufficient condition for a subset $E$ in the Heisenberg group (endowed with the Carnot-Carath\'{e}odory metric) to be contained in a rectifiable curve is that it satisfies a modified analogue of Peter Jones's geometric lemma. Our estimates improve on those of \cite{FFP}, by replacing the power $2$ of the Jones-$\beta$-number with any power $r<4$. This complements (in an open ended way) our work \cite{Li-Schul-beta-leq-length}, where we showed that such an estimate was necessary, but with $r=4$.