# Structure of Porous Sets in Carnot Groups

created by pinamonti on 15 Jul 2016
modified on 23 Jul 2017

[BibTeX]

Illinois Journal of Math.

Inserted: 15 jul 2016
Last Updated: 23 jul 2017

Year: 2016

ArXiv: 1607.04681 PDF

Abstract:

A set is porous if each point sees relatively large holes in the set on arbitrarily small scales. We show that sets porous with respect to the Carnot-Carath\'eodory distance are much smaller than measure zero sets and are not comparable with sets porous with respect to the Euclidean distance. We construct a Lipschitz function which is Pansu differentiable at no point of a given sigma-porous set and show preimages of open sets under the horizontal gradient are far from being porous.