{"paper":{"title":"A sharp necessary condition for rectifiable curves in metric spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Guy C. David, Raanan Schul","submitted_at":"2019-02-11T18:11:27Z","abstract_excerpt":"In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable curves in the plane, using a multiscale sum of what is now known as Jones $\\beta$-numbers, numbers measuring flatness in a given scale and location. This work was generalized to R^n by Okikiolu, to Hilbert space by the second author, and has many variants in a variety of metric settings. Notably, in 2005, Hahlomaa gave a sufficient condition for a subset of a metric space to be contained in a rectifiable curve. We prove the sharpest possible converse to Hahlomaa's theorem for doubling curves, and then deduce some coro"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.04030","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"}