{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:7RMLW3ASPM24MIC2BJI6SLMI6O","short_pith_number":"pith:7RMLW3AS","canonical_record":{"source":{"id":"1301.0096","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-01-01T15:35:06Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"38bfb792e8af92cd91db79c04207cb129dccb0324e07723330febdb5b9f5cb58","abstract_canon_sha256":"c1c83fb899f3e162163b78dc146d29526898d115619a05a5b1ffc789c717ebee"},"schema_version":"1.0"},"canonical_sha256":"fc58bb6c127b35c6205a0a51e92d88f39236dab10e4a245bd21c5a349ba6d51c","source":{"kind":"arxiv","id":"1301.0096","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1301.0096","created_at":"2026-05-18T03:13:58Z"},{"alias_kind":"arxiv_version","alias_value":"1301.0096v2","created_at":"2026-05-18T03:13:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1301.0096","created_at":"2026-05-18T03:13:58Z"},{"alias_kind":"pith_short_12","alias_value":"7RMLW3ASPM24","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_16","alias_value":"7RMLW3ASPM24MIC2","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_8","alias_value":"7RMLW3AS","created_at":"2026-05-18T12:27:36Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:7RMLW3ASPM24MIC2BJI6SLMI6O","target":"record","payload":{"canonical_record":{"source":{"id":"1301.0096","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-01-01T15:35:06Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"38bfb792e8af92cd91db79c04207cb129dccb0324e07723330febdb5b9f5cb58","abstract_canon_sha256":"c1c83fb899f3e162163b78dc146d29526898d115619a05a5b1ffc789c717ebee"},"schema_version":"1.0"},"canonical_sha256":"fc58bb6c127b35c6205a0a51e92d88f39236dab10e4a245bd21c5a349ba6d51c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:13:58.271593Z","signature_b64":"iNQK421WekjXVPfg5geI5tAvpVdb2dKKwBaWhxc6SRaFfrYSWFkJqYLxIt+o0bn0o+fy8FOm7xuQyfAP5dxmAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fc58bb6c127b35c6205a0a51e92d88f39236dab10e4a245bd21c5a349ba6d51c","last_reissued_at":"2026-05-18T03:13:58.271042Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:13:58.271042Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1301.0096","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":"2026-05-18T03:13:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wgJFSL9TxtC9oe6rPdjW05zoQUtAeYyIF9og9Hn0b1KmKZNNoRzClUHfDy7z2DFLU6Ao7N8aUO4laJU4XofDCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T06:12:01.777812Z"},"content_sha256":"f435cc0a1c30edf374da408a1335874d4a0d27eb8690bbfb8177134658f64ff7","schema_version":"1.0","event_id":"sha256:f435cc0a1c30edf374da408a1335874d4a0d27eb8690bbfb8177134658f64ff7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:7RMLW3ASPM24MIC2BJI6SLMI6O","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Structure of Critical Product Sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.CO","authors_text":"Matt DeVos","submitted_at":"2013-01-01T15:35:06Z","abstract_excerpt":"Let $G$ be a multiplicative group, let $A,B \\subseteq G$ be finite and nonempty, and define the product set $AB = {ab \\mid $a \\in A$ and $b \\in B$}$. Two fundamental problems in combinatorial number theory are to find lower bounds on $|AB|$, and then to determine structural properties of $A$ and $B$ under the assumption that $|AB|$ is small. We focus on the extreme case when $|AB| < |A| + |B|$, and call any such pair $(A,B)$ \\emph{critical}.\n  In the case when $|G|$ is prime, the Cauchy-Davenport Theorem asserts that $|AB| \\ge \\min {|G|, |A| + |B| - 1}$, and Vosper refined this result by class"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.0096","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":"2026-05-18T03:13:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kt3QHsqHFd/M+3rfk0V5AkH3Jor/UGtEatQcerFh15G5qBMc5rm4Nwqqz+ik9Y2ac8jfEGRkqoDCqgDEDQdADw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T06:12:01.778418Z"},"content_sha256":"5a64e03af6fd4413281e5a1520ae561ec419c3613a0cb2acc12b15ae3c08c654","schema_version":"1.0","event_id":"sha256:5a64e03af6fd4413281e5a1520ae561ec419c3613a0cb2acc12b15ae3c08c654"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7RMLW3ASPM24MIC2BJI6SLMI6O/bundle.json","state_url":"https://pith.science/pith/7RMLW3ASPM24MIC2BJI6SLMI6O/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7RMLW3ASPM24MIC2BJI6SLMI6O/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":"2026-05-30T06:12:01Z","links":{"resolver":"https://pith.science/pith/7RMLW3ASPM24MIC2BJI6SLMI6O","bundle":"https://pith.science/pith/7RMLW3ASPM24MIC2BJI6SLMI6O/bundle.json","state":"https://pith.science/pith/7RMLW3ASPM24MIC2BJI6SLMI6O/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7RMLW3ASPM24MIC2BJI6SLMI6O/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:7RMLW3ASPM24MIC2BJI6SLMI6O","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":"c1c83fb899f3e162163b78dc146d29526898d115619a05a5b1ffc789c717ebee","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-01-01T15:35:06Z","title_canon_sha256":"38bfb792e8af92cd91db79c04207cb129dccb0324e07723330febdb5b9f5cb58"},"schema_version":"1.0","source":{"id":"1301.0096","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1301.0096","created_at":"2026-05-18T03:13:58Z"},{"alias_kind":"arxiv_version","alias_value":"1301.0096v2","created_at":"2026-05-18T03:13:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1301.0096","created_at":"2026-05-18T03:13:58Z"},{"alias_kind":"pith_short_12","alias_value":"7RMLW3ASPM24","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_16","alias_value":"7RMLW3ASPM24MIC2","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_8","alias_value":"7RMLW3AS","created_at":"2026-05-18T12:27:36Z"}],"graph_snapshots":[{"event_id":"sha256:5a64e03af6fd4413281e5a1520ae561ec419c3613a0cb2acc12b15ae3c08c654","target":"graph","created_at":"2026-05-18T03:13:58Z","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"},"paper":{"abstract_excerpt":"Let $G$ be a multiplicative group, let $A,B \\subseteq G$ be finite and nonempty, and define the product set $AB = {ab \\mid $a \\in A$ and $b \\in B$}$. Two fundamental problems in combinatorial number theory are to find lower bounds on $|AB|$, and then to determine structural properties of $A$ and $B$ under the assumption that $|AB|$ is small. We focus on the extreme case when $|AB| < |A| + |B|$, and call any such pair $(A,B)$ \\emph{critical}.\n  In the case when $|G|$ is prime, the Cauchy-Davenport Theorem asserts that $|AB| \\ge \\min {|G|, |A| + |B| - 1}$, and Vosper refined this result by class","authors_text":"Matt DeVos","cross_cats":["math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-01-01T15:35:06Z","title":"The Structure of Critical Product Sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.0096","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:f435cc0a1c30edf374da408a1335874d4a0d27eb8690bbfb8177134658f64ff7","target":"record","created_at":"2026-05-18T03:13:58Z","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":"c1c83fb899f3e162163b78dc146d29526898d115619a05a5b1ffc789c717ebee","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-01-01T15:35:06Z","title_canon_sha256":"38bfb792e8af92cd91db79c04207cb129dccb0324e07723330febdb5b9f5cb58"},"schema_version":"1.0","source":{"id":"1301.0096","kind":"arxiv","version":2}},"canonical_sha256":"fc58bb6c127b35c6205a0a51e92d88f39236dab10e4a245bd21c5a349ba6d51c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fc58bb6c127b35c6205a0a51e92d88f39236dab10e4a245bd21c5a349ba6d51c","first_computed_at":"2026-05-18T03:13:58.271042Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:13:58.271042Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"iNQK421WekjXVPfg5geI5tAvpVdb2dKKwBaWhxc6SRaFfrYSWFkJqYLxIt+o0bn0o+fy8FOm7xuQyfAP5dxmAg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:13:58.271593Z","signed_message":"canonical_sha256_bytes"},"source_id":"1301.0096","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f435cc0a1c30edf374da408a1335874d4a0d27eb8690bbfb8177134658f64ff7","sha256:5a64e03af6fd4413281e5a1520ae561ec419c3613a0cb2acc12b15ae3c08c654"],"state_sha256":"8c080aa79df0097a8dc98c066bae5098614b3e4cc22612a2531dcb5e070a7ca9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qc8rmYrt+dWxfrs0a8HSEbwFFcFfPDrvpDPtTp2+fNv6rc06wZrg6yH2uxyLRpml7npTpZQ6Y3IQc4fAyIZKBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T06:12:01.780587Z","bundle_sha256":"486c045df08ac0ee849aaf50cc6b2cde87faddf038ff91390ae96cd953af35a2"}}