{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:VFC6E3KAPYX7SLFECCMBIJFZSR","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":"81d3f00792c53238026fd79cde5f41946dd251d02ebe08b94cf47ddb52f0bc1a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-07-19T19:59:27Z","title_canon_sha256":"df2ca48fff97ed026b119a39af11904b08722b20e928284b931c7aa554938076"},"schema_version":"1.0","source":{"id":"1607.05718","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1607.05718","created_at":"2026-05-18T01:10:46Z"},{"alias_kind":"arxiv_version","alias_value":"1607.05718v1","created_at":"2026-05-18T01:10:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.05718","created_at":"2026-05-18T01:10:46Z"},{"alias_kind":"pith_short_12","alias_value":"VFC6E3KAPYX7","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_16","alias_value":"VFC6E3KAPYX7SLFE","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_8","alias_value":"VFC6E3KA","created_at":"2026-05-18T12:30:48Z"}],"graph_snapshots":[{"event_id":"sha256:31f6b896346aca25281ec4ef0a7112e666231dbd361d8d386a8daf8e8b058f0d","target":"graph","created_at":"2026-05-18T01:10:46Z","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 an abelian group of finite order $n$, and let $h$ be a positive integer. A subset $A$ of $G$ is called {\\em weakly $h$-incomplete}, if not every element of $G$ can be written as the sum of $h$ distinct elements of $A$; in particular, if $A$ does not contain $h$ distinct elements that add to zero, then $A$ is called {\\em weakly $h$-zero-sum-free}. We investigate the maximum size of weakly $h$-incomplete and weakly $h$-zero-sum-free sets in $G$, denoted by $C_h(G)$ and $Z_h(G)$, respectively. Among our results are the following: (i) If $G$ is of odd order and $(n-1)/2 \\leq h \\leq n-2$","authors_text":"B\\'ela Bajnok, Samuel Edwards","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-07-19T19:59:27Z","title":"On two questions about restricted sumsets in finite abelian groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.05718","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:ffd87f572dfb4b3df17cc535a45bd75d7c0f4ccb309ac569384538cbddb939d3","target":"record","created_at":"2026-05-18T01:10:46Z","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":"81d3f00792c53238026fd79cde5f41946dd251d02ebe08b94cf47ddb52f0bc1a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-07-19T19:59:27Z","title_canon_sha256":"df2ca48fff97ed026b119a39af11904b08722b20e928284b931c7aa554938076"},"schema_version":"1.0","source":{"id":"1607.05718","kind":"arxiv","version":1}},"canonical_sha256":"a945e26d407e2ff92ca410981424b9944836415bd270c628d57e964060694c44","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a945e26d407e2ff92ca410981424b9944836415bd270c628d57e964060694c44","first_computed_at":"2026-05-18T01:10:46.191217Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:10:46.191217Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ZFjiBqbEQOChc2SwsvezhXS9EgAgdxIEm/BM7P6uV6tPxEOYh6nWkY/wA46iR/gOi+WMNgcPNQuZZG0EbUjfDw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:10:46.191615Z","signed_message":"canonical_sha256_bytes"},"source_id":"1607.05718","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ffd87f572dfb4b3df17cc535a45bd75d7c0f4ccb309ac569384538cbddb939d3","sha256:31f6b896346aca25281ec4ef0a7112e666231dbd361d8d386a8daf8e8b058f0d"],"state_sha256":"5744aab1a49452b3e89d9167d58ea8afef70512917a8ebe01343225e9da1689b"}