{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:BEJZ4MIS2FX7IRFFEPDPUWNU6V","short_pith_number":"pith:BEJZ4MIS","canonical_record":{"source":{"id":"1810.08127","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-10-18T16:01:45Z","cross_cats_sorted":["math.CO","math.MG"],"title_canon_sha256":"6a1167228342382a6a76be66a686f1d27291cbf34c23e4804f9775519703651b","abstract_canon_sha256":"96f81e71d6860f11eac8562c396f0b2ecddfde914b8dd60c52cc370f5b134acc"},"schema_version":"1.0"},"canonical_sha256":"09139e3112d16ff444a523c6fa59b4f5598be7f384d801d50706f8100925e94a","source":{"kind":"arxiv","id":"1810.08127","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1810.08127","created_at":"2026-07-05T00:16:58Z"},{"alias_kind":"arxiv_version","alias_value":"1810.08127v3","created_at":"2026-07-05T00:16:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.08127","created_at":"2026-07-05T00:16:58Z"},{"alias_kind":"pith_short_12","alias_value":"BEJZ4MIS2FX7","created_at":"2026-07-05T00:16:58Z"},{"alias_kind":"pith_short_16","alias_value":"BEJZ4MIS2FX7IRFF","created_at":"2026-07-05T00:16:58Z"},{"alias_kind":"pith_short_8","alias_value":"BEJZ4MIS","created_at":"2026-07-05T00:16:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:BEJZ4MIS2FX7IRFFEPDPUWNU6V","target":"record","payload":{"canonical_record":{"source":{"id":"1810.08127","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-10-18T16:01:45Z","cross_cats_sorted":["math.CO","math.MG"],"title_canon_sha256":"6a1167228342382a6a76be66a686f1d27291cbf34c23e4804f9775519703651b","abstract_canon_sha256":"96f81e71d6860f11eac8562c396f0b2ecddfde914b8dd60c52cc370f5b134acc"},"schema_version":"1.0"},"canonical_sha256":"09139e3112d16ff444a523c6fa59b4f5598be7f384d801d50706f8100925e94a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:16:58.835833Z","signature_b64":"++9/Q8CEmoHnc+mBbbowqpIt/i4xb7tRX+7LkTmZYF/ra5c3YxsiRPndfqn6eMUwt8uXNEF1LR2tvfgZ098NAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"09139e3112d16ff444a523c6fa59b4f5598be7f384d801d50706f8100925e94a","last_reissued_at":"2026-07-05T00:16:58.835457Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:16:58.835457Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1810.08127","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T00:16:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZyPiRV0iOw90SO8z71YjY7mYPzyuOfnZPy4/gHBUOw1wLQxG0RFucFcDY/0apSl4PnENVzij4M0PjOGmVZmEDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-18T03:27:57.493779Z"},"content_sha256":"652130e531d77d3e1519f5a60750c976cbd95ea6f8ef8c0531fba22878e0bfe5","schema_version":"1.0","event_id":"sha256:652130e531d77d3e1519f5a60750c976cbd95ea6f8ef8c0531fba22878e0bfe5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:BEJZ4MIS2FX7IRFFEPDPUWNU6V","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Hausdorff dimension of pinned distance sets and the $L^2$-method","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":"2018-10-18T16:01:45Z","abstract_excerpt":"We prove that for any $E\\subset{\\Bbb R}^2$, $\\dim_{\\mathcal{H}}(E)>1$, there exists $x\\in E$ such that the Hausdorff dimension of the pinned distance set\n  $$\\Delta_x(E)=\\{|x-y|: y \\in E\\}$$\n  is no less than $\\min\\left\\{\\frac{4}{3}\\dim_{\\mathcal{H}}(E)-\\frac{2}{3}, 1\\right\\}$. This answers a question recently raised by Guth, Iosevich, Ou and Wang, as well as improves results of Keleti and Shmerkin.\n  (This version is already published on Proceeding AMS so I would like to leave it unchanged. However the statement in the abstract, which is the second part of Theorem 1.1, should be weakened a bi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.08127","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1810.08127/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T00:16:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3k0q4jJAcP8h/RjP3kQe7H3CxJ3Q5aphpPmtCXZCHOTGpqxmp8UX7Mku7eFoYbl2+rR6x9j9vHsj4aI59/SeAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-18T03:27:57.494737Z"},"content_sha256":"8b9cd082f5213045f150053b984729bbfa7baad65b95c43de8babc8413f5c5bc","schema_version":"1.0","event_id":"sha256:8b9cd082f5213045f150053b984729bbfa7baad65b95c43de8babc8413f5c5bc"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BEJZ4MIS2FX7IRFFEPDPUWNU6V/bundle.json","state_url":"https://pith.science/pith/BEJZ4MIS2FX7IRFFEPDPUWNU6V/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BEJZ4MIS2FX7IRFFEPDPUWNU6V/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-18T03:27:57Z","links":{"resolver":"https://pith.science/pith/BEJZ4MIS2FX7IRFFEPDPUWNU6V","bundle":"https://pith.science/pith/BEJZ4MIS2FX7IRFFEPDPUWNU6V/bundle.json","state":"https://pith.science/pith/BEJZ4MIS2FX7IRFFEPDPUWNU6V/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BEJZ4MIS2FX7IRFFEPDPUWNU6V/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:BEJZ4MIS2FX7IRFFEPDPUWNU6V","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":"96f81e71d6860f11eac8562c396f0b2ecddfde914b8dd60c52cc370f5b134acc","cross_cats_sorted":["math.CO","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-10-18T16:01:45Z","title_canon_sha256":"6a1167228342382a6a76be66a686f1d27291cbf34c23e4804f9775519703651b"},"schema_version":"1.0","source":{"id":"1810.08127","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1810.08127","created_at":"2026-07-05T00:16:58Z"},{"alias_kind":"arxiv_version","alias_value":"1810.08127v3","created_at":"2026-07-05T00:16:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.08127","created_at":"2026-07-05T00:16:58Z"},{"alias_kind":"pith_short_12","alias_value":"BEJZ4MIS2FX7","created_at":"2026-07-05T00:16:58Z"},{"alias_kind":"pith_short_16","alias_value":"BEJZ4MIS2FX7IRFF","created_at":"2026-07-05T00:16:58Z"},{"alias_kind":"pith_short_8","alias_value":"BEJZ4MIS","created_at":"2026-07-05T00:16:58Z"}],"graph_snapshots":[{"event_id":"sha256:8b9cd082f5213045f150053b984729bbfa7baad65b95c43de8babc8413f5c5bc","target":"graph","created_at":"2026-07-05T00:16:58Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1810.08127/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove that for any $E\\subset{\\Bbb R}^2$, $\\dim_{\\mathcal{H}}(E)>1$, there exists $x\\in E$ such that the Hausdorff dimension of the pinned distance set\n  $$\\Delta_x(E)=\\{|x-y|: y \\in E\\}$$\n  is no less than $\\min\\left\\{\\frac{4}{3}\\dim_{\\mathcal{H}}(E)-\\frac{2}{3}, 1\\right\\}$. This answers a question recently raised by Guth, Iosevich, Ou and Wang, as well as improves results of Keleti and Shmerkin.\n  (This version is already published on Proceeding AMS so I would like to leave it unchanged. However the statement in the abstract, which is the second part of Theorem 1.1, should be weakened a bi","authors_text":"Bochen Liu","cross_cats":["math.CO","math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-10-18T16:01:45Z","title":"Hausdorff dimension of pinned distance sets and the $L^2$-method"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.08127","kind":"arxiv","version":3},"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":"sha256:652130e531d77d3e1519f5a60750c976cbd95ea6f8ef8c0531fba22878e0bfe5","target":"record","created_at":"2026-07-05T00:16:58Z","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":"96f81e71d6860f11eac8562c396f0b2ecddfde914b8dd60c52cc370f5b134acc","cross_cats_sorted":["math.CO","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-10-18T16:01:45Z","title_canon_sha256":"6a1167228342382a6a76be66a686f1d27291cbf34c23e4804f9775519703651b"},"schema_version":"1.0","source":{"id":"1810.08127","kind":"arxiv","version":3}},"canonical_sha256":"09139e3112d16ff444a523c6fa59b4f5598be7f384d801d50706f8100925e94a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"09139e3112d16ff444a523c6fa59b4f5598be7f384d801d50706f8100925e94a","first_computed_at":"2026-07-05T00:16:58.835457Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T00:16:58.835457Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"++9/Q8CEmoHnc+mBbbowqpIt/i4xb7tRX+7LkTmZYF/ra5c3YxsiRPndfqn6eMUwt8uXNEF1LR2tvfgZ098NAw==","signature_status":"signed_v1","signed_at":"2026-07-05T00:16:58.835833Z","signed_message":"canonical_sha256_bytes"},"source_id":"1810.08127","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:652130e531d77d3e1519f5a60750c976cbd95ea6f8ef8c0531fba22878e0bfe5","sha256:8b9cd082f5213045f150053b984729bbfa7baad65b95c43de8babc8413f5c5bc"],"state_sha256":"70a5326b0d1a359e193019712e018b2f18da7c4df835b7fd93c0d91d63ae91b8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1oggL6dSeuEs90J2lHt/zt2c8jNB21UIQ/9nXCxYsCJPN3/Bpfc//7r3D1IiTWhbOLc4mjWMY8bejux2LJEcAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-18T03:27:57.517482Z","bundle_sha256":"1f286b4a4b9170320a25bd26080e92560ca2996ad9e6be5f8a085a7cd0889b6c"}}