{"paper":{"title":"Structure of Porous Sets in Carnot Groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.MG","authors_text":"Andrea Pinamonti, Gareth Speight","submitted_at":"2016-07-15T23:18:01Z","abstract_excerpt":"We show that any Carnot group contains a closed nowhere dense set which has measure zero but is not $\\sigma$-porous with respect to the Carnot-Carath\\'eodory (CC) distance. In the first Heisenberg group we observe that there exist sets which are porous with respect to the CC distance but not the Euclidean distance and vice-versa. In Carnot groups we then 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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.04681","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}