{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:YGPMD4YDFP4XTP4FHVUGFGITVT","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":"45264d4fddc31ee5fac598e76cec4bffb5080f81381886ee551a5a852a5abf9d","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-10-31T01:55:13Z","title_canon_sha256":"a3297e350a7a8cacc9c5c3f718f3d3a676920a08900f2a03f2b54c7c144abb9f"},"schema_version":"1.0","source":{"id":"1410.8614","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.8614","created_at":"2026-05-18T02:38:54Z"},{"alias_kind":"arxiv_version","alias_value":"1410.8614v1","created_at":"2026-05-18T02:38:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.8614","created_at":"2026-05-18T02:38:54Z"},{"alias_kind":"pith_short_12","alias_value":"YGPMD4YDFP4X","created_at":"2026-05-18T12:28:57Z"},{"alias_kind":"pith_short_16","alias_value":"YGPMD4YDFP4XTP4F","created_at":"2026-05-18T12:28:57Z"},{"alias_kind":"pith_short_8","alias_value":"YGPMD4YD","created_at":"2026-05-18T12:28:57Z"}],"graph_snapshots":[{"event_id":"sha256:397f5f61cad26d47cb254058cf6bfd2a03f7b962ce39b6b871ce17ee91da43f1","target":"graph","created_at":"2026-05-18T02:38:54Z","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 $d\\geq 2$, $A \\subset \\mathbb{Z}^d$ be finite and not contained in a translate of any hyperplane, and $q \\in \\mathbb{Z}$ such that $|q| > 1$. We show $$|A+ q \\cdot A| \\geq (|q|+d+1)|A| - O_{q,d}(1).$$","authors_text":"Antal Balog, George Shakan","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-10-31T01:55:13Z","title":"Sum of dilates in vector spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.8614","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":"sha256:5a3d535c95375acb08db7a3669d9d8bbec9aae57d37ad1469c3bc43b1dec4c7d","target":"record","created_at":"2026-05-18T02:38:54Z","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":"45264d4fddc31ee5fac598e76cec4bffb5080f81381886ee551a5a852a5abf9d","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-10-31T01:55:13Z","title_canon_sha256":"a3297e350a7a8cacc9c5c3f718f3d3a676920a08900f2a03f2b54c7c144abb9f"},"schema_version":"1.0","source":{"id":"1410.8614","kind":"arxiv","version":1}},"canonical_sha256":"c19ec1f3032bf979bf853d68629913acf8e72ddb4992ec7467266d32d9892443","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c19ec1f3032bf979bf853d68629913acf8e72ddb4992ec7467266d32d9892443","first_computed_at":"2026-05-18T02:38:54.116645Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:38:54.116645Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Ph4CJKt57ZUTx/nY8XDLZTgnFtJAXZMBt7xCommiX7KVQ3s1cAsmznzNxuLZbnixBUHrNw7JAjcVFRBKj6xgCA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:38:54.117189Z","signed_message":"canonical_sha256_bytes"},"source_id":"1410.8614","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5a3d535c95375acb08db7a3669d9d8bbec9aae57d37ad1469c3bc43b1dec4c7d","sha256:397f5f61cad26d47cb254058cf6bfd2a03f7b962ce39b6b871ce17ee91da43f1"],"state_sha256":"fccfe6623657469867aa3a11162183182676c39d72f367a226c39613f3d213d7"}