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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1604.04091","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-04-14T09:52:36Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"86ea4d8470622faca4b45fd0faf26313428a21d65cde50a522381fe7894190fd","abstract_canon_sha256":"79ebc7f39e3ffb6c6ff9ce6c6512ee141a866e8fef555e6ed168bbe4a7fd933d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:08:31.667787Z","signature_b64":"ntrWqNVU8jW3hoXGNXxZvFRcs+D9VDn1SfhiPobh0EAmFf3AYDupZdfg2g1s9Z3AlzbI95REt/Gk9reKAAllBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"14f719577b44ec2e23397b0563a5268fb996f97330195fcd7109637ae3d0a3d9","last_reissued_at":"2026-05-18T00:08:31.667144Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:08:31.667144Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1604.04091","created_at":"2026-05-18T00:08:31.667276+00:00"},{"alias_kind":"arxiv_version","alias_value":"1604.04091v1","created_at":"2026-05-18T00:08:31.667276+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.04091","created_at":"2026-05-18T00:08:31.667276+00:00"},{"alias_kind":"pith_short_12","alias_value":"CT3RSV33ITWC","created_at":"2026-05-18T12:30:09.641336+00:00"},{"alias_kind":"pith_short_16","alias_value":"CT3RSV33ITWC4IZZ","created_at":"2026-05-18T12:30:09.641336+00:00"},{"alias_kind":"pith_short_8","alias_value":"CT3RSV33","created_at":"2026-05-18T12:30:09.641336+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/CT3RSV33ITWC4IZZPMCWHJJGR6","json":"https://pith.science/pith/CT3RSV33ITWC4IZZPMCWHJJGR6.json","graph_json":"https://pith.science/api/pith-number/CT3RSV33ITWC4IZZPMCWHJJGR6/graph.json","events_json":"https://pith.science/api/pith-number/CT3RSV33ITWC4IZZPMCWHJJGR6/events.json","paper":"https://pith.science/paper/CT3RSV33"},"agent_actions":{"view_html":"https://pith.science/pith/CT3RSV33ITWC4IZZPMCWHJJGR6","download_json":"https://pith.science/pith/CT3RSV33ITWC4IZZPMCWHJJGR6.json","view_paper":"https://pith.science/paper/CT3RSV33","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1604.04091&json=true","fetch_graph":"https://pith.science/api/pith-number/CT3RSV33ITWC4IZZPMCWHJJGR6/graph.json","fetch_events":"https://pith.science/api/pith-number/CT3RSV33ITWC4IZZPMCWHJJGR6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CT3RSV33ITWC4IZZPMCWHJJGR6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CT3RSV33ITWC4IZZPMCWHJJGR6/action/storage_attestation","attest_author":"https://pith.science/pith/CT3RSV33ITWC4IZZPMCWHJJGR6/action/author_attestation","sign_citation":"https://pith.science/pith/CT3RSV33ITWC4IZZPMCWHJJGR6/action/citation_signature","submit_replication":"https://pith.science/pith/CT3RSV33ITWC4IZZPMCWHJJGR6/action/replication_record"}},"created_at":"2026-05-18T00:08:31.667276+00:00","updated_at":"2026-05-18T00:08:31.667276+00:00"}