{"paper":{"title":"Multiscale analysis of 1-rectifiable measures: necessary conditions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CA","authors_text":"Matthew Badger, Raanan Schul","submitted_at":"2013-07-02T19:39:17Z","abstract_excerpt":"We repurpose tools from the theory of quantitative rectifiability to study the qualitative rectifiability of measures in $\\Bbb{R}^n$, $n\\geq 2$. To each locally finite Borel measure $\\mu$, we associate a function $\\widetilde J_2(\\mu, x)$ which uses a weighted sum to record how closely the mass of $\\mu$ is concentrated on a line in the triples of dyadic cubes containing $x$. We show that $\\widetilde J_2(\\mu, x) < \\infty$ $\\mu$-a.e. is a necessary condition for $\\mu$ to give full mass to a countable family of rectifiable curves. This confirms a conjecture of Peter Jones from 2000. A novelty of t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.0804","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"}