{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:2G4RO35ZB3YVBR7UQDZG4WIKZ2","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"0da925b5590ee6714125246060838c17b7b4db5b9cd40db06704b9e5500c3cee","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-10-31T14:31:14Z","title_canon_sha256":"45a325b81da1f9b93cc0a90cd1daa777e3bfd430b4565d578fb3211788ffe073"},"schema_version":"1.0","source":{"id":"1110.6805","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.6805","created_at":"2026-05-18T04:09:54Z"},{"alias_kind":"arxiv_version","alias_value":"1110.6805v1","created_at":"2026-05-18T04:09:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.6805","created_at":"2026-05-18T04:09:54Z"},{"alias_kind":"pith_short_12","alias_value":"2G4RO35ZB3YV","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_16","alias_value":"2G4RO35ZB3YVBR7U","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_8","alias_value":"2G4RO35Z","created_at":"2026-05-18T12:26:18Z"}],"graph_snapshots":[{"event_id":"sha256:9e5e6275cf1850c2abe6b0a1e0960bd5de6d6379fba578eef83859e9ef30dca8","target":"graph","created_at":"2026-05-18T04:09:54Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We extend a result, due to Mattila and Sjolin, which says that if the Hausdorff dimension of a compact set $E \\subset {\\Bbb R}^d$, $d \\ge 2$, is greater than $\\frac{d+1}{2}$, then the distance set $\\Delta(E)=\\{|x-y|: x,y \\in E \\}$ contains an interval. We prove this result for distance sets $\\Delta_B(E)=\\{{||x-y||}_B: x,y \\in E \\}$, where ${|| \\cdot ||}_B$ is the metric induced by the norm defined by a symmetric bounded convex body $B$ with a smooth boundary and everywhere non-vanishing Gaussian curvature. We also obtain some detailed estimates pertaining to the Radon-Nikodym derivative of the","authors_text":"Alex Iosevich, Krystal Taylor, Mihalis Mourgoglou","cross_cats":["math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-10-31T14:31:14Z","title":"On the Mattila-Sjolin theorem for distance sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.6805","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:36ebdf6d5165bad6cc2ddbfe6d7583ac4bb632e6453c33339b08e43f14975f1e","target":"record","created_at":"2026-05-18T04:09:54Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"0da925b5590ee6714125246060838c17b7b4db5b9cd40db06704b9e5500c3cee","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-10-31T14:31:14Z","title_canon_sha256":"45a325b81da1f9b93cc0a90cd1daa777e3bfd430b4565d578fb3211788ffe073"},"schema_version":"1.0","source":{"id":"1110.6805","kind":"arxiv","version":1}},"canonical_sha256":"d1b9176fb90ef150c7f480f26e590ace8097ae3d86f1fff17d264bb03566c750","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d1b9176fb90ef150c7f480f26e590ace8097ae3d86f1fff17d264bb03566c750","first_computed_at":"2026-05-18T04:09:54.154157Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:09:54.154157Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"iV7R+kDrSn/IFVRGLIPxGybIhWr9ACQ1dYiEjqUV1i58ZfAvJ+ah71v3B/jVJRtfcOKk7R1oK864u5qq+rWmDA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:09:54.154656Z","signed_message":"canonical_sha256_bytes"},"source_id":"1110.6805","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:36ebdf6d5165bad6cc2ddbfe6d7583ac4bb632e6453c33339b08e43f14975f1e","sha256:9e5e6275cf1850c2abe6b0a1e0960bd5de6d6379fba578eef83859e9ef30dca8"],"state_sha256":"b3aac8b4002e2e48279316da07f8b8b9bef570e166d164389bd68ff75dc7fe3a"}