{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:KXEQAK6JYVHHQ2TWZBRDO6A7Y4","short_pith_number":"pith:KXEQAK6J","canonical_record":{"source":{"id":"1108.4500","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-08-23T05:46:04Z","cross_cats_sorted":[],"title_canon_sha256":"9612b17b70ccac80c2ef786d663f989b281410672d0d7998792259b8b0f7382a","abstract_canon_sha256":"9c82713da2581561f4b339c435121a45e7ee044335cabc8de94631794305960f"},"schema_version":"1.0"},"canonical_sha256":"55c9002bc9c54e786a76c86237781fc71840f7a392603e486044521d7936a631","source":{"kind":"arxiv","id":"1108.4500","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1108.4500","created_at":"2026-05-18T04:06:20Z"},{"alias_kind":"arxiv_version","alias_value":"1108.4500v3","created_at":"2026-05-18T04:06:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.4500","created_at":"2026-05-18T04:06:20Z"},{"alias_kind":"pith_short_12","alias_value":"KXEQAK6JYVHH","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_16","alias_value":"KXEQAK6JYVHHQ2TW","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_8","alias_value":"KXEQAK6J","created_at":"2026-05-18T12:26:34Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:KXEQAK6JYVHHQ2TWZBRDO6A7Y4","target":"record","payload":{"canonical_record":{"source":{"id":"1108.4500","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-08-23T05:46:04Z","cross_cats_sorted":[],"title_canon_sha256":"9612b17b70ccac80c2ef786d663f989b281410672d0d7998792259b8b0f7382a","abstract_canon_sha256":"9c82713da2581561f4b339c435121a45e7ee044335cabc8de94631794305960f"},"schema_version":"1.0"},"canonical_sha256":"55c9002bc9c54e786a76c86237781fc71840f7a392603e486044521d7936a631","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:06:20.653968Z","signature_b64":"F3Tcr7cVkxowews5t1hs4UjMRQjvdkmLYtTeedNQ3fgmFhtIXqCfiaS9GMEEqqBXyoqUCC192MHHrHfaXieWDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"55c9002bc9c54e786a76c86237781fc71840f7a392603e486044521d7936a631","last_reissued_at":"2026-05-18T04:06:20.653177Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:06:20.653177Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1108.4500","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":"2026-05-18T04:06:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mnHk4xFTtMh8VqNZfCO5h0vdQY6kGLSp+w7aIVgbtziqL127cbjFVvpIW17Pwo17QzkRFc7HPBxP9A4Bg3h9DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T16:20:59.644798Z"},"content_sha256":"9f149a04b36ae9753555e29a91d64182b840076505e07dc8d6c28533e4462cd7","schema_version":"1.0","event_id":"sha256:9f149a04b36ae9753555e29a91d64182b840076505e07dc8d6c28533e4462cd7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:KXEQAK6JYVHHQ2TWZBRDO6A7Y4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Generalized More Sums Than Differences Sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Geoffrey Iyer, Liyang Zhang, Oleg Lazarev, Steven J. Miller","submitted_at":"2011-08-23T05:46:04Z","abstract_excerpt":"A More Sums Than Differences (MSTD, or sum-dominant) set is a finite set $A\\subset \\mathbb{Z}$ such that $|A+A|<|A-A|$. Though it was believed that the percentage of subsets of $\\{0,...,n\\}$ that are sum-dominant tends to zero, in 2006 Martin and O'Bryant \\cite{MO} proved a positive percentage are sum-dominant. We generalize their result to the many different ways of taking sums and differences of a set. We prove that $|\\epsilon_1A+...+\\epsilon_kA|>|\\delta_1A+...+\\delta_kA|$ a positive percent of the time for all nontrivial choices of $\\epsilon_j,\\delta_j\\in \\{-1,1\\}$. Previous approaches prov"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.4500","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":""},"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":"2026-05-18T04:06:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gN9fJopq/3nXLiDHslpeOmtGKon4PrNarW+vRl7Su4fnxi+zSZbbIHL7PKawvl3HA/S6u1Dug8LgRhrd9QzeDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T16:20:59.645136Z"},"content_sha256":"b9616e10a302c69ebbe9f1b6df5d48761cf179fcb2224afb7284326539952d5b","schema_version":"1.0","event_id":"sha256:b9616e10a302c69ebbe9f1b6df5d48761cf179fcb2224afb7284326539952d5b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KXEQAK6JYVHHQ2TWZBRDO6A7Y4/bundle.json","state_url":"https://pith.science/pith/KXEQAK6JYVHHQ2TWZBRDO6A7Y4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KXEQAK6JYVHHQ2TWZBRDO6A7Y4/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":"2026-06-03T16:20:59Z","links":{"resolver":"https://pith.science/pith/KXEQAK6JYVHHQ2TWZBRDO6A7Y4","bundle":"https://pith.science/pith/KXEQAK6JYVHHQ2TWZBRDO6A7Y4/bundle.json","state":"https://pith.science/pith/KXEQAK6JYVHHQ2TWZBRDO6A7Y4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KXEQAK6JYVHHQ2TWZBRDO6A7Y4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:KXEQAK6JYVHHQ2TWZBRDO6A7Y4","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":"9c82713da2581561f4b339c435121a45e7ee044335cabc8de94631794305960f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-08-23T05:46:04Z","title_canon_sha256":"9612b17b70ccac80c2ef786d663f989b281410672d0d7998792259b8b0f7382a"},"schema_version":"1.0","source":{"id":"1108.4500","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1108.4500","created_at":"2026-05-18T04:06:20Z"},{"alias_kind":"arxiv_version","alias_value":"1108.4500v3","created_at":"2026-05-18T04:06:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.4500","created_at":"2026-05-18T04:06:20Z"},{"alias_kind":"pith_short_12","alias_value":"KXEQAK6JYVHH","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_16","alias_value":"KXEQAK6JYVHHQ2TW","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_8","alias_value":"KXEQAK6J","created_at":"2026-05-18T12:26:34Z"}],"graph_snapshots":[{"event_id":"sha256:b9616e10a302c69ebbe9f1b6df5d48761cf179fcb2224afb7284326539952d5b","target":"graph","created_at":"2026-05-18T04:06:20Z","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":"A More Sums Than Differences (MSTD, or sum-dominant) set is a finite set $A\\subset \\mathbb{Z}$ such that $|A+A|<|A-A|$. Though it was believed that the percentage of subsets of $\\{0,...,n\\}$ that are sum-dominant tends to zero, in 2006 Martin and O'Bryant \\cite{MO} proved a positive percentage are sum-dominant. We generalize their result to the many different ways of taking sums and differences of a set. We prove that $|\\epsilon_1A+...+\\epsilon_kA|>|\\delta_1A+...+\\delta_kA|$ a positive percent of the time for all nontrivial choices of $\\epsilon_j,\\delta_j\\in \\{-1,1\\}$. Previous approaches prov","authors_text":"Geoffrey Iyer, Liyang Zhang, Oleg Lazarev, Steven J. Miller","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-08-23T05:46:04Z","title":"Generalized More Sums Than Differences Sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.4500","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":"sha256:9f149a04b36ae9753555e29a91d64182b840076505e07dc8d6c28533e4462cd7","target":"record","created_at":"2026-05-18T04:06:20Z","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":"9c82713da2581561f4b339c435121a45e7ee044335cabc8de94631794305960f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-08-23T05:46:04Z","title_canon_sha256":"9612b17b70ccac80c2ef786d663f989b281410672d0d7998792259b8b0f7382a"},"schema_version":"1.0","source":{"id":"1108.4500","kind":"arxiv","version":3}},"canonical_sha256":"55c9002bc9c54e786a76c86237781fc71840f7a392603e486044521d7936a631","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"55c9002bc9c54e786a76c86237781fc71840f7a392603e486044521d7936a631","first_computed_at":"2026-05-18T04:06:20.653177Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:06:20.653177Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"F3Tcr7cVkxowews5t1hs4UjMRQjvdkmLYtTeedNQ3fgmFhtIXqCfiaS9GMEEqqBXyoqUCC192MHHrHfaXieWDA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:06:20.653968Z","signed_message":"canonical_sha256_bytes"},"source_id":"1108.4500","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9f149a04b36ae9753555e29a91d64182b840076505e07dc8d6c28533e4462cd7","sha256:b9616e10a302c69ebbe9f1b6df5d48761cf179fcb2224afb7284326539952d5b"],"state_sha256":"e808a7cae5eb4199272a970128b1d0085607c21cf41c6409bb9ab3d6215cc68d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ebnCj1vfqMKtDz9p76U4uN/M67L6kujX4w0/5u5Nolfs5O+acfSXrOXKCwnCp0t8G9gjojwE6m9tRXbC/fTMDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T16:20:59.647059Z","bundle_sha256":"36ff98318a4f262d04deef3f067e1b4f45f4a5a50cd6c856c0e4cd7b5992b682"}}