{"paper":{"title":"Boundedness of the density normalised Jones' square function does not imply $1$-rectifiability","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CA","authors_text":"Henri Martikainen, Tuomas Orponen","submitted_at":"2016-04-14T09:52:36Z","abstract_excerpt":"Recently, M. Badger and R. Schul proved that for a $1$-rectifiable Radon measure $\\mu$, the density weighted Jones' square function $$ J_{1}(x) = \\mathop{\\sum_{Q \\in \\mathcal{D}}}_{\\ell(Q) \\leq 1} \\beta_{2,\\mu}^{2}(3Q)\\frac{\\ell(Q)}{\\mu(Q)} 1_{Q}(x) $$ is finite for $\\mu$-a.e. $x$. Answering a question of Badger-Schul, we show that the converse is not true. Given $\\epsilon > 0$, we construct a Radon probability measure on $[0,1]^{2} \\subset \\mathbb{R}^{2}$ with the properties that $J_{1}(x) \\leq \\epsilon$ for all $x \\in \\operatorname{spt} \\mu$, but nevertheless the $1$-dimensional lower densit"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.04091","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"}