Remarks about Besicovitch covering property in Carnot groups of step 3 and higher

TL;DR

This paper demonstrates that the Besicovitch Covering Property fails for certain homogeneous quasi-distances in Carnot groups of step 3 and higher, contrasting with the case of Heisenberg groups where it holds.

Contribution

It establishes the non-validity of BCP for specific homogeneous quasi-distances in higher-step Carnot groups, extending understanding beyond the Heisenberg case.

Findings

BCP does not hold for some homogeneous quasi-distances in Carnot groups of step 3 and higher.

Homogeneous distances with Euclidean unit balls centered at the origin do not satisfy BCP in these groups.

Contrast with Heisenberg groups where such distances satisfy BCP.

Abstract

We prove that the Besicovitch Covering Property (BCP) does not hold for some classes of homogeneous quasi-distances on Carnot groups of step 3 and higher. As a special case we get that, in Carnot groups of step 3 and higher, BCP is not satisfied for those homogeneous distances whose unit ball centered at the origin coincides with a Euclidean ball centered at the origin. This result comes in constrast with the case of the Heisenberg groups where such distances satisfy BCP.