{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2022:EC2E2KBQUGPCFAGGH7ZS6SRLDF","short_pith_number":"pith:EC2E2KBQ","canonical_record":{"source":{"id":"2203.15731","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.MG","submitted_at":"2022-03-29T16:41:41Z","cross_cats_sorted":["math.CA","math.NT"],"title_canon_sha256":"752ddfcc15046c53dde239fb04a7df6f072b15a48a3d4c987f1b3b9568313d9d","abstract_canon_sha256":"84600b4ff894c121c097f364e686e36a4874b6c65a1932a9fe52fbe09ebbb603"},"schema_version":"1.0"},"canonical_sha256":"20b44d2830a19e2280c63ff32f4a2b194deaf17fd4b49113a3fab6f92f6eee7e","source":{"kind":"arxiv","id":"2203.15731","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2203.15731","created_at":"2026-07-05T06:31:03Z"},{"alias_kind":"arxiv_version","alias_value":"2203.15731v3","created_at":"2026-07-05T06:31:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2203.15731","created_at":"2026-07-05T06:31:03Z"},{"alias_kind":"pith_short_12","alias_value":"EC2E2KBQUGPC","created_at":"2026-07-05T06:31:03Z"},{"alias_kind":"pith_short_16","alias_value":"EC2E2KBQUGPCFAGG","created_at":"2026-07-05T06:31:03Z"},{"alias_kind":"pith_short_8","alias_value":"EC2E2KBQ","created_at":"2026-07-05T06:31:03Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2022:EC2E2KBQUGPCFAGGH7ZS6SRLDF","target":"record","payload":{"canonical_record":{"source":{"id":"2203.15731","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.MG","submitted_at":"2022-03-29T16:41:41Z","cross_cats_sorted":["math.CA","math.NT"],"title_canon_sha256":"752ddfcc15046c53dde239fb04a7df6f072b15a48a3d4c987f1b3b9568313d9d","abstract_canon_sha256":"84600b4ff894c121c097f364e686e36a4874b6c65a1932a9fe52fbe09ebbb603"},"schema_version":"1.0"},"canonical_sha256":"20b44d2830a19e2280c63ff32f4a2b194deaf17fd4b49113a3fab6f92f6eee7e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:31:03.504265Z","signature_b64":"xePhda7C0DUwTL0dM2PSbL0uBOM3j0PxnS1p4uH7XrrKOnLfsH0qotObbPMP7sQqqSKfu6K/2wvhJWMCDm2VBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"20b44d2830a19e2280c63ff32f4a2b194deaf17fd4b49113a3fab6f92f6eee7e","last_reissued_at":"2026-07-05T06:31:03.503798Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:31:03.503798Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2203.15731","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":"2026-07-05T06:31:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yed2ayjXk08DhTI7l03O37oOPv98QUDCPScoiJalmdYpgjAMCJUuoAQYE4hhhpalG6IVgs5tuJREeldJQfjJBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T16:30:42.784673Z"},"content_sha256":"9407ffed0a959d0217adae3ec98907083285b0a2302692f059b33e579c6288a3","schema_version":"1.0","event_id":"sha256:9407ffed0a959d0217adae3ec98907083285b0a2302692f059b33e579c6288a3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2022:EC2E2KBQUGPCFAGGH7ZS6SRLDF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Equivalences between different forms of the Kakeya conjecture and duality of Hausdorff and packing dimensions for additive complements","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CA","math.NT"],"primary_cat":"math.MG","authors_text":"Andr\\'as M\\'ath\\'e, Tam\\'as Keleti","submitted_at":"2022-03-29T16:41:41Z","abstract_excerpt":"The Kakeya conjecture is generally formulated as one the following statements: every compact/Borel/arbitrary subset of ${\\mathbb R}^n$ that contains a (unit) line segment in every direction has Hausdorff dimension $n$; or, sometimes, that every closed/Borel/arbitrary subset of ${\\mathbb R}^n$ that contains a full line in every direction has Hausdorff dimension $n$. These statements are generally expected to be equivalent. Moreover, the condition that the set contains a line (segment) in every direction is often relaxed by requiring a line (segment) for a \"large\" set of directions only, where l"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2203.15731","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/2203.15731/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":"2026-07-05T06:31:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mj4G9O+slZbeweXy4eSjzlUSAhD37a54mx+0n41+IY1wZfe3zd1X2mtHwFkjFOrG+SGup6UNkzFOmju6ZnnQBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T16:30:42.785581Z"},"content_sha256":"3435ee9a1fa94becd475112ead0719dd8eff9790c266201e61a0e9754c714346","schema_version":"1.0","event_id":"sha256:3435ee9a1fa94becd475112ead0719dd8eff9790c266201e61a0e9754c714346"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EC2E2KBQUGPCFAGGH7ZS6SRLDF/bundle.json","state_url":"https://pith.science/pith/EC2E2KBQUGPCFAGGH7ZS6SRLDF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EC2E2KBQUGPCFAGGH7ZS6SRLDF/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":"2026-08-12T16:30:42Z","links":{"resolver":"https://pith.science/pith/EC2E2KBQUGPCFAGGH7ZS6SRLDF","bundle":"https://pith.science/pith/EC2E2KBQUGPCFAGGH7ZS6SRLDF/bundle.json","state":"https://pith.science/pith/EC2E2KBQUGPCFAGGH7ZS6SRLDF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EC2E2KBQUGPCFAGGH7ZS6SRLDF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:EC2E2KBQUGPCFAGGH7ZS6SRLDF","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":"84600b4ff894c121c097f364e686e36a4874b6c65a1932a9fe52fbe09ebbb603","cross_cats_sorted":["math.CA","math.NT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.MG","submitted_at":"2022-03-29T16:41:41Z","title_canon_sha256":"752ddfcc15046c53dde239fb04a7df6f072b15a48a3d4c987f1b3b9568313d9d"},"schema_version":"1.0","source":{"id":"2203.15731","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2203.15731","created_at":"2026-07-05T06:31:03Z"},{"alias_kind":"arxiv_version","alias_value":"2203.15731v3","created_at":"2026-07-05T06:31:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2203.15731","created_at":"2026-07-05T06:31:03Z"},{"alias_kind":"pith_short_12","alias_value":"EC2E2KBQUGPC","created_at":"2026-07-05T06:31:03Z"},{"alias_kind":"pith_short_16","alias_value":"EC2E2KBQUGPCFAGG","created_at":"2026-07-05T06:31:03Z"},{"alias_kind":"pith_short_8","alias_value":"EC2E2KBQ","created_at":"2026-07-05T06:31:03Z"}],"graph_snapshots":[{"event_id":"sha256:3435ee9a1fa94becd475112ead0719dd8eff9790c266201e61a0e9754c714346","target":"graph","created_at":"2026-07-05T06:31:03Z","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/2203.15731/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The Kakeya conjecture is generally formulated as one the following statements: every compact/Borel/arbitrary subset of ${\\mathbb R}^n$ that contains a (unit) line segment in every direction has Hausdorff dimension $n$; or, sometimes, that every closed/Borel/arbitrary subset of ${\\mathbb R}^n$ that contains a full line in every direction has Hausdorff dimension $n$. These statements are generally expected to be equivalent. Moreover, the condition that the set contains a line (segment) in every direction is often relaxed by requiring a line (segment) for a \"large\" set of directions only, where l","authors_text":"Andr\\'as M\\'ath\\'e, Tam\\'as Keleti","cross_cats":["math.CA","math.NT"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.MG","submitted_at":"2022-03-29T16:41:41Z","title":"Equivalences between different forms of the Kakeya conjecture and duality of Hausdorff and packing dimensions for additive complements"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2203.15731","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":"sha256:9407ffed0a959d0217adae3ec98907083285b0a2302692f059b33e579c6288a3","target":"record","created_at":"2026-07-05T06:31:03Z","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":"84600b4ff894c121c097f364e686e36a4874b6c65a1932a9fe52fbe09ebbb603","cross_cats_sorted":["math.CA","math.NT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.MG","submitted_at":"2022-03-29T16:41:41Z","title_canon_sha256":"752ddfcc15046c53dde239fb04a7df6f072b15a48a3d4c987f1b3b9568313d9d"},"schema_version":"1.0","source":{"id":"2203.15731","kind":"arxiv","version":3}},"canonical_sha256":"20b44d2830a19e2280c63ff32f4a2b194deaf17fd4b49113a3fab6f92f6eee7e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"20b44d2830a19e2280c63ff32f4a2b194deaf17fd4b49113a3fab6f92f6eee7e","first_computed_at":"2026-07-05T06:31:03.503798Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T06:31:03.503798Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xePhda7C0DUwTL0dM2PSbL0uBOM3j0PxnS1p4uH7XrrKOnLfsH0qotObbPMP7sQqqSKfu6K/2wvhJWMCDm2VBA==","signature_status":"signed_v1","signed_at":"2026-07-05T06:31:03.504265Z","signed_message":"canonical_sha256_bytes"},"source_id":"2203.15731","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9407ffed0a959d0217adae3ec98907083285b0a2302692f059b33e579c6288a3","sha256:3435ee9a1fa94becd475112ead0719dd8eff9790c266201e61a0e9754c714346"],"state_sha256":"26843d477b628e0c17920b8f52848a86bd9e1a40ae7ee47cee31682f9e2ba066"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SgVq2Cq2f7Xz82ntPijIYJgwI/+xZtFlkFjx9zqe2N88YFyvnoy8Ix18r7GUoWt1SX6xvy/dcyfrBIPfTkIUDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-12T16:30:42.791304Z","bundle_sha256":"4f8e2063dc05db68ba6235e6336684d93a62cfa630032366c65f17ace4c8a6d9"}}