{"paper":{"title":"Packing dimension and Ahlfors regularity of porous sets in metric spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Antti K\\\"aenm\\\"aki, Esa J\\\"arvenp\\\"a\\\"a, Maarit J\\\"arvenp\\\"a\\\"a, Sari Rogovin, Tapio Rajala, Ville Suomala","submitted_at":"2017-01-30T13:48:21Z","abstract_excerpt":"Let $X$ be a metric measure space with an $s$-regular measure $\\mu$. We prove that if $A\\subset X$ is $\\varrho$-porous, then $\\dim_{\\mathrm{p}}(A)\\le s-c\\varrho^s$ where $\\dim_{\\mathrm{p}}$ is the packing dimension and $c$ is a positive constant which depends on $s$ and the structure constants of $\\mu$. This is an analogue of a well known asymptotically sharp result in Euclidean spaces. We illustrate by an example that the corresponding result is not valid if $\\mu$ is a doubling measure. However, in the doubling case we find a fixed $N\\subset X$ with $\\mu(N)=0$ such that $\\dim_{\\mathrm{p}}(A)\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.08593","kind":"arxiv","version":1},"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"}