An upper bound for the length of a traveling salesman path in the Heisenberg group

Abstract

We show that a sufficient condition for a subset $E$ in the Heisenberg group (endowed with the Carnot–Carathéodory 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 [6], allowing for any power $r$ < 4 to replace the power 2 of the Jones-$\beta$-number. This complements in an open ended way our work in [13], where we showed that such an estimate was necessary, however with the power $r$ = 4.