{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:HRY3265Q34IMWB5TANQS2BYCX2","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":"4c086fb9941f7329b0dc28ae2ba5d88a1161c6b3aa51fe72944dbe8a7672ab0e","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-01-16T09:36:07Z","title_canon_sha256":"4e06f355eaf8face32fb5e3fb65be431aa7b038c6d574089a4dbbfc9c8640546"},"schema_version":"1.0","source":{"id":"1601.04146","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.04146","created_at":"2026-05-18T01:22:45Z"},{"alias_kind":"arxiv_version","alias_value":"1601.04146v1","created_at":"2026-05-18T01:22:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.04146","created_at":"2026-05-18T01:22:45Z"},{"alias_kind":"pith_short_12","alias_value":"HRY3265Q34IM","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_16","alias_value":"HRY3265Q34IMWB5T","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_8","alias_value":"HRY3265Q","created_at":"2026-05-18T12:30:19Z"}],"graph_snapshots":[{"event_id":"sha256:cda1e2d5bf0440277807db9052ee76d1be16517998c3fa7bc3587a23bbaaed73","target":"graph","created_at":"2026-05-18T01:22:45Z","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":"We compare the size of the difference set $A-A$ to that of the set $kA$ of $k$-fold sums. We show the existence of sets such that $|kA| < |A-A|^{a_k}$ with $a_k<1$.","authors_text":"Imre Z. Ruzsa","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-01-16T09:36:07Z","title":"More differences than multiple sums"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.04146","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":"sha256:85fac7cd1f4723ee6176ca0337e78ca63cacc4102f858b003b88e3179f8ddc8d","target":"record","created_at":"2026-05-18T01:22:45Z","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":"4c086fb9941f7329b0dc28ae2ba5d88a1161c6b3aa51fe72944dbe8a7672ab0e","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-01-16T09:36:07Z","title_canon_sha256":"4e06f355eaf8face32fb5e3fb65be431aa7b038c6d574089a4dbbfc9c8640546"},"schema_version":"1.0","source":{"id":"1601.04146","kind":"arxiv","version":1}},"canonical_sha256":"3c71bd7bb0df10cb07b303612d0702bebc71ca2f20efa9377056fffd7038c5f0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3c71bd7bb0df10cb07b303612d0702bebc71ca2f20efa9377056fffd7038c5f0","first_computed_at":"2026-05-18T01:22:45.775209Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:22:45.775209Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"KSwpy6WUB2tqyfuuB7XeLutMl3EeZkcXRb+rWLmQbWfybhFAHZNC3SkiZtsD6KuvQMeXdd+23aJ5/diN5rXzBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:22:45.775734Z","signed_message":"canonical_sha256_bytes"},"source_id":"1601.04146","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:85fac7cd1f4723ee6176ca0337e78ca63cacc4102f858b003b88e3179f8ddc8d","sha256:cda1e2d5bf0440277807db9052ee76d1be16517998c3fa7bc3587a23bbaaed73"],"state_sha256":"54cd7f208c5493b1dab04d06ed0e901a2ef9e12e129265febc16030a2af05aac"}