{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:A3PJDIIAJEACCKWXQZYK7TW7C5","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":"704c2a7cf1b212940911ed454bfbb4eacbba2c0bb8f46787f3a2f4e87b1a16e5","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-06-28T02:33:20Z","title_canon_sha256":"14fce89705e9d0763c61cfe4d1d70ae2d44f5512598bcabf6c754007d918a6ac"},"schema_version":"1.0","source":{"id":"1406.7346","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1406.7346","created_at":"2026-05-18T02:26:12Z"},{"alias_kind":"arxiv_version","alias_value":"1406.7346v1","created_at":"2026-05-18T02:26:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1406.7346","created_at":"2026-05-18T02:26:12Z"},{"alias_kind":"pith_short_12","alias_value":"A3PJDIIAJEAC","created_at":"2026-05-18T12:28:19Z"},{"alias_kind":"pith_short_16","alias_value":"A3PJDIIAJEACCKWX","created_at":"2026-05-18T12:28:19Z"},{"alias_kind":"pith_short_8","alias_value":"A3PJDIIA","created_at":"2026-05-18T12:28:19Z"}],"graph_snapshots":[{"event_id":"sha256:4ca18e2fe30a0041dae5f915b890d6c50e7bb6eebf658485bfc2f450fd74c4a6","target":"graph","created_at":"2026-05-18T02:26:12Z","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 $A=\\{a_0,a_1,\\ldots,a_{k-1}\\}$ be a set of $k$ integers. For any integer $h\\ge 1$ and any ordered $k$-tuple of positive integers $\\mathbf{r}=(r_0,r_1,\\ldots,r_{k-1})$, we define a general $h$-fold sumset, denoted by $h^{(\\mathbf{r})}A$, which is the set of all sums of $h$ elements of $A$, where $a_i$ appearing in the sum can be repeated at most $r_i$ times for $i=0,1,\\ldots,k-1$. In this paper, we give the best lower bound for $|h^{(\\mathbf{r})}A|$ in terms of $\\mathbf{r}$ and $h$ and determine the structure of the set $A$ when $|h^{(\\mathbf{r})}A|$ is minimal. This generalizes results of ","authors_text":"Quan-Hui Yang, Yong-Gao Chen","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-06-28T02:33:20Z","title":"On the cardinality of general $h$-fold sumsets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.7346","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":"sha256:0f0d1f0cd17cc8dc6e96abc7f626aca099d17fe4adec15374f13fee9e97defe7","target":"record","created_at":"2026-05-18T02:26:12Z","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":"704c2a7cf1b212940911ed454bfbb4eacbba2c0bb8f46787f3a2f4e87b1a16e5","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-06-28T02:33:20Z","title_canon_sha256":"14fce89705e9d0763c61cfe4d1d70ae2d44f5512598bcabf6c754007d918a6ac"},"schema_version":"1.0","source":{"id":"1406.7346","kind":"arxiv","version":1}},"canonical_sha256":"06de91a1004900212ad78670afcedf17631af9e112e8b62b175d79ec41fd5935","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"06de91a1004900212ad78670afcedf17631af9e112e8b62b175d79ec41fd5935","first_computed_at":"2026-05-18T02:26:12.805199Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:26:12.805199Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"FI97/o12RGbHw9wWtFrJNY4MqljG650QzvVI56jg36EfJuIWXyxmuqsw6yT7MMRZRsWcu7Vislnc9qcfJL5oCg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:26:12.805815Z","signed_message":"canonical_sha256_bytes"},"source_id":"1406.7346","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0f0d1f0cd17cc8dc6e96abc7f626aca099d17fe4adec15374f13fee9e97defe7","sha256:4ca18e2fe30a0041dae5f915b890d6c50e7bb6eebf658485bfc2f450fd74c4a6"],"state_sha256":"75772710b04231cc9a24172caa7891c9e9a80d2f1bff2fc40b09605939020114"}