{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:Z6YXJ5BICED3TT33Q265WXVJZY","short_pith_number":"pith:Z6YXJ5BI","canonical_record":{"source":{"id":"1405.3132","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-05-13T12:53:41Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"9a37241f7f746700aa0a8d57628c185615bac0252b6f74a9f6e729c03d49db5a","abstract_canon_sha256":"c710438645e6b3ecde59e321d7a02aeb3af8d3e0652ac808367ffdcfabbca12d"},"schema_version":"1.0"},"canonical_sha256":"cfb174f4281107b9cf7b86bddb5ea9ce3ec784ee617639e645cfe1b47252ee81","source":{"kind":"arxiv","id":"1405.3132","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1405.3132","created_at":"2026-05-18T02:51:57Z"},{"alias_kind":"arxiv_version","alias_value":"1405.3132v1","created_at":"2026-05-18T02:51:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1405.3132","created_at":"2026-05-18T02:51:57Z"},{"alias_kind":"pith_short_12","alias_value":"Z6YXJ5BICED3","created_at":"2026-05-18T12:28:59Z"},{"alias_kind":"pith_short_16","alias_value":"Z6YXJ5BICED3TT33","created_at":"2026-05-18T12:28:59Z"},{"alias_kind":"pith_short_8","alias_value":"Z6YXJ5BI","created_at":"2026-05-18T12:28:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:Z6YXJ5BICED3TT33Q265WXVJZY","target":"record","payload":{"canonical_record":{"source":{"id":"1405.3132","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-05-13T12:53:41Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"9a37241f7f746700aa0a8d57628c185615bac0252b6f74a9f6e729c03d49db5a","abstract_canon_sha256":"c710438645e6b3ecde59e321d7a02aeb3af8d3e0652ac808367ffdcfabbca12d"},"schema_version":"1.0"},"canonical_sha256":"cfb174f4281107b9cf7b86bddb5ea9ce3ec784ee617639e645cfe1b47252ee81","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:51:57.692478Z","signature_b64":"v8fJxw02i0ummVkqUpM0f1H2equeC7oVrdr6h24+NN0RuUzy7TvGWJ6BAFBjUSDIIoCjqfsHtGh0mpyDdMi8Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cfb174f4281107b9cf7b86bddb5ea9ce3ec784ee617639e645cfe1b47252ee81","last_reissued_at":"2026-05-18T02:51:57.691848Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:51:57.691848Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1405.3132","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:51:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"grE4YZcvyBBaCU2hLTBIqblbmbgHDLgDccs4hiXZ4fgZMlh5DD/2P6SImg6M+WPMyutBHA3eGj4Ca4gkxdpdBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T19:40:59.600662Z"},"content_sha256":"df890df13bf5f05260554ff511b622aea66bcdfd9f46a5267a2117e8a0938ee7","schema_version":"1.0","event_id":"sha256:df890df13bf5f05260554ff511b622aea66bcdfd9f46a5267a2117e8a0938ee7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:Z6YXJ5BICED3TT33Q265WXVJZY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Energies and structure of additive sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Ilya D. Shkredov","submitted_at":"2014-05-13T12:53:41Z","abstract_excerpt":"In the paper we prove that any sumset or difference set has large E_3 energy. Also, we give a full description of families of sets having critical relations between some kind of energies such as E_k, T_k and Gowers norms. In particular, we give criteria for a set to be a 1) set of the form H+L, where H+H is small and L has \"random structure\", 2) set equals a disjoint union of sets H_j, each H_j has small doubling, 3) set having large subset A' with 2A' is equal to a set with small doubling and |A'+A'| \\approx |A|^4 / \\E(A)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.3132","kind":"arxiv","version":1},"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":"2026-05-18T02:51:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2pbvW/fabehmJ+Z6KmQLQzchUnIqUR/X25NYKJtqO0xbCtvBQrGFceOHLwB1VDrMdlGY8dpxL+MX1Epvp+xQBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T19:40:59.601431Z"},"content_sha256":"0189d1cba6aae5a95774041618a2260e5eba1b6380f37fd8867b0d4c49a2088c","schema_version":"1.0","event_id":"sha256:0189d1cba6aae5a95774041618a2260e5eba1b6380f37fd8867b0d4c49a2088c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Z6YXJ5BICED3TT33Q265WXVJZY/bundle.json","state_url":"https://pith.science/pith/Z6YXJ5BICED3TT33Q265WXVJZY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Z6YXJ5BICED3TT33Q265WXVJZY/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":"2026-06-09T19:40:59Z","links":{"resolver":"https://pith.science/pith/Z6YXJ5BICED3TT33Q265WXVJZY","bundle":"https://pith.science/pith/Z6YXJ5BICED3TT33Q265WXVJZY/bundle.json","state":"https://pith.science/pith/Z6YXJ5BICED3TT33Q265WXVJZY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Z6YXJ5BICED3TT33Q265WXVJZY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:Z6YXJ5BICED3TT33Q265WXVJZY","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":"c710438645e6b3ecde59e321d7a02aeb3af8d3e0652ac808367ffdcfabbca12d","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-05-13T12:53:41Z","title_canon_sha256":"9a37241f7f746700aa0a8d57628c185615bac0252b6f74a9f6e729c03d49db5a"},"schema_version":"1.0","source":{"id":"1405.3132","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1405.3132","created_at":"2026-05-18T02:51:57Z"},{"alias_kind":"arxiv_version","alias_value":"1405.3132v1","created_at":"2026-05-18T02:51:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1405.3132","created_at":"2026-05-18T02:51:57Z"},{"alias_kind":"pith_short_12","alias_value":"Z6YXJ5BICED3","created_at":"2026-05-18T12:28:59Z"},{"alias_kind":"pith_short_16","alias_value":"Z6YXJ5BICED3TT33","created_at":"2026-05-18T12:28:59Z"},{"alias_kind":"pith_short_8","alias_value":"Z6YXJ5BI","created_at":"2026-05-18T12:28:59Z"}],"graph_snapshots":[{"event_id":"sha256:0189d1cba6aae5a95774041618a2260e5eba1b6380f37fd8867b0d4c49a2088c","target":"graph","created_at":"2026-05-18T02:51:57Z","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":"In the paper we prove that any sumset or difference set has large E_3 energy. Also, we give a full description of families of sets having critical relations between some kind of energies such as E_k, T_k and Gowers norms. In particular, we give criteria for a set to be a 1) set of the form H+L, where H+H is small and L has \"random structure\", 2) set equals a disjoint union of sets H_j, each H_j has small doubling, 3) set having large subset A' with 2A' is equal to a set with small doubling and |A'+A'| \\approx |A|^4 / \\E(A).","authors_text":"Ilya D. Shkredov","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-05-13T12:53:41Z","title":"Energies and structure of additive sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.3132","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:df890df13bf5f05260554ff511b622aea66bcdfd9f46a5267a2117e8a0938ee7","target":"record","created_at":"2026-05-18T02:51:57Z","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":"c710438645e6b3ecde59e321d7a02aeb3af8d3e0652ac808367ffdcfabbca12d","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-05-13T12:53:41Z","title_canon_sha256":"9a37241f7f746700aa0a8d57628c185615bac0252b6f74a9f6e729c03d49db5a"},"schema_version":"1.0","source":{"id":"1405.3132","kind":"arxiv","version":1}},"canonical_sha256":"cfb174f4281107b9cf7b86bddb5ea9ce3ec784ee617639e645cfe1b47252ee81","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cfb174f4281107b9cf7b86bddb5ea9ce3ec784ee617639e645cfe1b47252ee81","first_computed_at":"2026-05-18T02:51:57.691848Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:51:57.691848Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"v8fJxw02i0ummVkqUpM0f1H2equeC7oVrdr6h24+NN0RuUzy7TvGWJ6BAFBjUSDIIoCjqfsHtGh0mpyDdMi8Bg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:51:57.692478Z","signed_message":"canonical_sha256_bytes"},"source_id":"1405.3132","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:df890df13bf5f05260554ff511b622aea66bcdfd9f46a5267a2117e8a0938ee7","sha256:0189d1cba6aae5a95774041618a2260e5eba1b6380f37fd8867b0d4c49a2088c"],"state_sha256":"8501e70f57b26beb39908a1b514351e8934e2a5a41b1be79e7e4942ef77bb648"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"67ajmLP+PvQKeFkukJRgQ1gFox9urW/U24GnEZC3rk4ZisNxyW92EiLpLmm1bPEEuq/uSo0jnFGHK/UmxL4cCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T19:40:59.605698Z","bundle_sha256":"88707c47f146c30b5dff6789edbc20827ca89532434516d96d8e429c792c8f96"}}