{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:XB6752RM3JRSLQXLHFS2E36M3H","short_pith_number":"pith:XB6752RM","schema_version":"1.0","canonical_sha256":"b87dfeea2cda6325c2eb3965a26fccd9fc14b15f9fa42967188b6f261049d239","source":{"kind":"arxiv","id":"1709.06814","version":2},"attestation_state":"computed","paper":{"title":"Improvement on $2$-chains inside thin subsets of Euclidean spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.MG"],"primary_cat":"math.CA","authors_text":"Bochen Liu","submitted_at":"2017-09-20T11:10:21Z","abstract_excerpt":"We prove that if the Hausdorff dimension of $E\\subset\\mathbb{R}^d$, $d\\geq 2$ is greater than $\\frac{d}{2}+\\frac{1}{3}$, the set of gaps of $2$-chains inside $E$, $$\\Delta_2(E)=\\{(|x-y|, |y-z|): x, y, z\\in E \\}\\subset\\mathbb{R}^2$$ has positive Lebesgue measure. It generalizes Wolff-Erdogan's result on distances and improves a result of Bennett, Iosevich and Taylor on finite chains.\n  We also consider the similarity class of $2$-chains, $$S_2(E)=\\left\\{\\frac{t_1}{t_2}:(t_1,t_2)\\in\\Delta_2(E)\\right\\}=\\left\\{\\frac{|x-y|}{|y-z|}: x, y, z\\in E \\right\\}\\subset\\mathbb{R},$$ and show that $|S_2(E)|>0"},"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":"1709.06814","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-09-20T11:10:21Z","cross_cats_sorted":["math.CO","math.MG"],"title_canon_sha256":"c81f01e4709d7270c593375c6a99289d130bd4586c737b585e842e4494dfa9a5","abstract_canon_sha256":"80e079a32f7e31fd485468f1352490029bc041c60eb0c1fc7dd0dba97c29c889"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:32:00.902061Z","signature_b64":"IrYUMQmt+tlt/PJRne8XoK1K9DEukCNjcbTYMnBg/LOmuKg4dAdk9EVkcqvx3cghDrODMx0EzPrCqm+zRJZJDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b87dfeea2cda6325c2eb3965a26fccd9fc14b15f9fa42967188b6f261049d239","last_reissued_at":"2026-05-18T00:32:00.901495Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:32:00.901495Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Improvement on $2$-chains inside thin subsets of Euclidean spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.MG"],"primary_cat":"math.CA","authors_text":"Bochen Liu","submitted_at":"2017-09-20T11:10:21Z","abstract_excerpt":"We prove that if the Hausdorff dimension of $E\\subset\\mathbb{R}^d$, $d\\geq 2$ is greater than $\\frac{d}{2}+\\frac{1}{3}$, the set of gaps of $2$-chains inside $E$, $$\\Delta_2(E)=\\{(|x-y|, |y-z|): x, y, z\\in E \\}\\subset\\mathbb{R}^2$$ has positive Lebesgue measure. It generalizes Wolff-Erdogan's result on distances and improves a result of Bennett, Iosevich and Taylor on finite chains.\n  We also consider the similarity class of $2$-chains, $$S_2(E)=\\left\\{\\frac{t_1}{t_2}:(t_1,t_2)\\in\\Delta_2(E)\\right\\}=\\left\\{\\frac{|x-y|}{|y-z|}: x, y, z\\in E \\right\\}\\subset\\mathbb{R},$$ and show that $|S_2(E)|>0"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.06814","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1709.06814","created_at":"2026-05-18T00:32:00.901577+00:00"},{"alias_kind":"arxiv_version","alias_value":"1709.06814v2","created_at":"2026-05-18T00:32:00.901577+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.06814","created_at":"2026-05-18T00:32:00.901577+00:00"},{"alias_kind":"pith_short_12","alias_value":"XB6752RM3JRS","created_at":"2026-05-18T12:31:53.515858+00:00"},{"alias_kind":"pith_short_16","alias_value":"XB6752RM3JRSLQXL","created_at":"2026-05-18T12:31:53.515858+00:00"},{"alias_kind":"pith_short_8","alias_value":"XB6752RM","created_at":"2026-05-18T12:31:53.515858+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/XB6752RM3JRSLQXLHFS2E36M3H","json":"https://pith.science/pith/XB6752RM3JRSLQXLHFS2E36M3H.json","graph_json":"https://pith.science/api/pith-number/XB6752RM3JRSLQXLHFS2E36M3H/graph.json","events_json":"https://pith.science/api/pith-number/XB6752RM3JRSLQXLHFS2E36M3H/events.json","paper":"https://pith.science/paper/XB6752RM"},"agent_actions":{"view_html":"https://pith.science/pith/XB6752RM3JRSLQXLHFS2E36M3H","download_json":"https://pith.science/pith/XB6752RM3JRSLQXLHFS2E36M3H.json","view_paper":"https://pith.science/paper/XB6752RM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1709.06814&json=true","fetch_graph":"https://pith.science/api/pith-number/XB6752RM3JRSLQXLHFS2E36M3H/graph.json","fetch_events":"https://pith.science/api/pith-number/XB6752RM3JRSLQXLHFS2E36M3H/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XB6752RM3JRSLQXLHFS2E36M3H/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XB6752RM3JRSLQXLHFS2E36M3H/action/storage_attestation","attest_author":"https://pith.science/pith/XB6752RM3JRSLQXLHFS2E36M3H/action/author_attestation","sign_citation":"https://pith.science/pith/XB6752RM3JRSLQXLHFS2E36M3H/action/citation_signature","submit_replication":"https://pith.science/pith/XB6752RM3JRSLQXLHFS2E36M3H/action/replication_record"}},"created_at":"2026-05-18T00:32:00.901577+00:00","updated_at":"2026-05-18T00:32:00.901577+00:00"}