{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:WCV5WGN2XPUZIY7HXTCZGAB57N","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":"27c41f731a50d94873c9b2363801c1d8c01bdb0202dbb634f7e53861f0a363d7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-02-05T10:11:37Z","title_canon_sha256":"add008a474c7d8db75dd1ad461abfc7edd55d59a57085b680c6c448017406521"},"schema_version":"1.0","source":{"id":"1502.01487","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1502.01487","created_at":"2026-05-18T02:27:51Z"},{"alias_kind":"arxiv_version","alias_value":"1502.01487v1","created_at":"2026-05-18T02:27:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.01487","created_at":"2026-05-18T02:27:51Z"},{"alias_kind":"pith_short_12","alias_value":"WCV5WGN2XPUZ","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_16","alias_value":"WCV5WGN2XPUZIY7H","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_8","alias_value":"WCV5WGN2","created_at":"2026-05-18T12:29:47Z"}],"graph_snapshots":[{"event_id":"sha256:b7f00cb2801d92dcc56f0d8bc5b0b24c42b7b61671186ac0140f3f7c97ec4245","target":"graph","created_at":"2026-05-18T02:27:51Z","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 study subsets of $\\R^{d}$ which are thin for doubling measures or isotropic doubling measures. We show that any subset of $\\R^{d}$ with Hausdorff dimension less than or equal to $d-1$ is thin for isotropic doubling measures. We also prove that a self-affine set that satisfies $OSCH$ (open set condition with holes) is thin for isotropic doubling measures. For doubling measures, we prove that Bara\\'nski carpets are thin for doubling measures.","authors_text":"Changhao Chen, Shengyou Wen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-02-05T10:11:37Z","title":"On thin carpets for doubling measures"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.01487","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":"sha256:17e470512b408af5be0bfa50a22abe85eabd2514ea5182d938c67c04ef478910","target":"record","created_at":"2026-05-18T02:27:51Z","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":"27c41f731a50d94873c9b2363801c1d8c01bdb0202dbb634f7e53861f0a363d7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-02-05T10:11:37Z","title_canon_sha256":"add008a474c7d8db75dd1ad461abfc7edd55d59a57085b680c6c448017406521"},"schema_version":"1.0","source":{"id":"1502.01487","kind":"arxiv","version":1}},"canonical_sha256":"b0abdb19babbe99463e7bcc593003dfb5031c7df2515133d1e0487d80ce446a1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b0abdb19babbe99463e7bcc593003dfb5031c7df2515133d1e0487d80ce446a1","first_computed_at":"2026-05-18T02:27:51.946393Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:27:51.946393Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"P0rxi6YTNQVVIRJLzwaXFzNz79QsSBmtODAGroQ3begb5rktuU2HPRJEMk6iu6KlsTkTXbcVfkUwiOafq0MmAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:27:51.947002Z","signed_message":"canonical_sha256_bytes"},"source_id":"1502.01487","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:17e470512b408af5be0bfa50a22abe85eabd2514ea5182d938c67c04ef478910","sha256:b7f00cb2801d92dcc56f0d8bc5b0b24c42b7b61671186ac0140f3f7c97ec4245"],"state_sha256":"372ac7c78c2567ef5e1a44e63b9bddb87c60e0886e20736d6ca68774408a56dc"}