{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:LLRHKBP7MRO7DYAJM5WXUV3I5J","short_pith_number":"pith:LLRHKBP7","canonical_record":{"source":{"id":"1003.5649","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-03-29T19:42:43Z","cross_cats_sorted":[],"title_canon_sha256":"43418949f862431ab657f453c92901ad74b80fb2132b551f197e7a9870100c18","abstract_canon_sha256":"c85b304b38f45bf1a2a3e1ce5efd04527d15e9f43d917a2d464e934f53256c5b"},"schema_version":"1.0"},"canonical_sha256":"5ae27505ff645df1e009676d7a5768ea7453fcc3a224981e5975ef9f563b0d25","source":{"kind":"arxiv","id":"1003.5649","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1003.5649","created_at":"2026-05-18T04:34:18Z"},{"alias_kind":"arxiv_version","alias_value":"1003.5649v2","created_at":"2026-05-18T04:34:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1003.5649","created_at":"2026-05-18T04:34:18Z"},{"alias_kind":"pith_short_12","alias_value":"LLRHKBP7MRO7","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_16","alias_value":"LLRHKBP7MRO7DYAJ","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_8","alias_value":"LLRHKBP7","created_at":"2026-05-18T12:26:10Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:LLRHKBP7MRO7DYAJM5WXUV3I5J","target":"record","payload":{"canonical_record":{"source":{"id":"1003.5649","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-03-29T19:42:43Z","cross_cats_sorted":[],"title_canon_sha256":"43418949f862431ab657f453c92901ad74b80fb2132b551f197e7a9870100c18","abstract_canon_sha256":"c85b304b38f45bf1a2a3e1ce5efd04527d15e9f43d917a2d464e934f53256c5b"},"schema_version":"1.0"},"canonical_sha256":"5ae27505ff645df1e009676d7a5768ea7453fcc3a224981e5975ef9f563b0d25","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:34:18.355836Z","signature_b64":"g8S4jSZw88icbPXYB53W73dmQjj52a5WlOAV1o9SAy23wmX321IoHsk5ogNL1EX64ujeuZWqDcjkCyHXr2oRDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5ae27505ff645df1e009676d7a5768ea7453fcc3a224981e5975ef9f563b0d25","last_reissued_at":"2026-05-18T04:34:18.355340Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:34:18.355340Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1003.5649","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:34:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"40WKbCYgz5gjdHMNNyyuiVrNSzAbboNbgKZOt31LFCJl0+3hdvrTSSpyVw6MSAzn/hWmuNRd1k5OVdoHfKaQDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T16:04:14.789012Z"},"content_sha256":"51e54c55026050aadf34ff468b80dda7122b6c681aa2c3fbe81a32647c36f79e","schema_version":"1.0","event_id":"sha256:51e54c55026050aadf34ff468b80dda7122b6c681aa2c3fbe81a32647c36f79e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:LLRHKBP7MRO7DYAJM5WXUV3I5J","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Green's sumset problem at density one half","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Tom Sanders","submitted_at":"2010-03-29T19:42:43Z","abstract_excerpt":"We investigate the size of subspaces in sumsets and show two main results. First, if A is a subset of F_2^n with density at least 1/2 - o(n^{-1/2}) then A+A contains a subspace of co-dimension 1. Secondly, if A is a subset of F_2^n with density at least 1/2-o(1) then A+A contains a subspace of co-dimension o(n)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1003.5649","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:34:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"d0Lk1a/5HDt7ah7VJLMLwzbQoSl1HMkBNJIuYaRxz3uWpdkbxDM67lRE2lu7cHyo9H84d6lTpjVV9Zlu82oQBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T16:04:14.789708Z"},"content_sha256":"0ef5b0c0d5485e6c0c8dc4357f97409ee7c70d7382182156a263784e40eb0421","schema_version":"1.0","event_id":"sha256:0ef5b0c0d5485e6c0c8dc4357f97409ee7c70d7382182156a263784e40eb0421"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LLRHKBP7MRO7DYAJM5WXUV3I5J/bundle.json","state_url":"https://pith.science/pith/LLRHKBP7MRO7DYAJM5WXUV3I5J/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LLRHKBP7MRO7DYAJM5WXUV3I5J/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":"2026-05-31T16:04:14Z","links":{"resolver":"https://pith.science/pith/LLRHKBP7MRO7DYAJM5WXUV3I5J","bundle":"https://pith.science/pith/LLRHKBP7MRO7DYAJM5WXUV3I5J/bundle.json","state":"https://pith.science/pith/LLRHKBP7MRO7DYAJM5WXUV3I5J/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LLRHKBP7MRO7DYAJM5WXUV3I5J/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:LLRHKBP7MRO7DYAJM5WXUV3I5J","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":"c85b304b38f45bf1a2a3e1ce5efd04527d15e9f43d917a2d464e934f53256c5b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-03-29T19:42:43Z","title_canon_sha256":"43418949f862431ab657f453c92901ad74b80fb2132b551f197e7a9870100c18"},"schema_version":"1.0","source":{"id":"1003.5649","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1003.5649","created_at":"2026-05-18T04:34:18Z"},{"alias_kind":"arxiv_version","alias_value":"1003.5649v2","created_at":"2026-05-18T04:34:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1003.5649","created_at":"2026-05-18T04:34:18Z"},{"alias_kind":"pith_short_12","alias_value":"LLRHKBP7MRO7","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_16","alias_value":"LLRHKBP7MRO7DYAJ","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_8","alias_value":"LLRHKBP7","created_at":"2026-05-18T12:26:10Z"}],"graph_snapshots":[{"event_id":"sha256:0ef5b0c0d5485e6c0c8dc4357f97409ee7c70d7382182156a263784e40eb0421","target":"graph","created_at":"2026-05-18T04:34:18Z","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 investigate the size of subspaces in sumsets and show two main results. First, if A is a subset of F_2^n with density at least 1/2 - o(n^{-1/2}) then A+A contains a subspace of co-dimension 1. Secondly, if A is a subset of F_2^n with density at least 1/2-o(1) then A+A contains a subspace of co-dimension o(n).","authors_text":"Tom Sanders","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-03-29T19:42:43Z","title":"Green's sumset problem at density one half"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1003.5649","kind":"arxiv","version":2},"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":"sha256:51e54c55026050aadf34ff468b80dda7122b6c681aa2c3fbe81a32647c36f79e","target":"record","created_at":"2026-05-18T04:34:18Z","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":"c85b304b38f45bf1a2a3e1ce5efd04527d15e9f43d917a2d464e934f53256c5b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-03-29T19:42:43Z","title_canon_sha256":"43418949f862431ab657f453c92901ad74b80fb2132b551f197e7a9870100c18"},"schema_version":"1.0","source":{"id":"1003.5649","kind":"arxiv","version":2}},"canonical_sha256":"5ae27505ff645df1e009676d7a5768ea7453fcc3a224981e5975ef9f563b0d25","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5ae27505ff645df1e009676d7a5768ea7453fcc3a224981e5975ef9f563b0d25","first_computed_at":"2026-05-18T04:34:18.355340Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:34:18.355340Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"g8S4jSZw88icbPXYB53W73dmQjj52a5WlOAV1o9SAy23wmX321IoHsk5ogNL1EX64ujeuZWqDcjkCyHXr2oRDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:34:18.355836Z","signed_message":"canonical_sha256_bytes"},"source_id":"1003.5649","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:51e54c55026050aadf34ff468b80dda7122b6c681aa2c3fbe81a32647c36f79e","sha256:0ef5b0c0d5485e6c0c8dc4357f97409ee7c70d7382182156a263784e40eb0421"],"state_sha256":"49595bf9026f377df2a5cde488f8fc98a6e1a86c219d2bc80eaeefa22cacc07f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Pbxx2L9zZtEXLC6Nao7mS+ibOekij1jJAgZj3SkDM9r5YbgNHC6ZKon2vZRlvWF2p7/2gD2l+choYASIiabyAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T16:04:14.793254Z","bundle_sha256":"51ee65e49c172cb5dfcd7e77a65bc92297e0acaa3e6eaff4c7d17fcce0351b41"}}