{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:AO3QNX27RII2X54HXQXXX7EZTT","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":"bfe7c661bf7bdfaec7ee6b4b26e39cb6f3e221ed72a945c3fe86d1f9d9dc54c2","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-01-26T09:47:10Z","title_canon_sha256":"08d6a89481b65b14c6a3a04a7cd6d7478d04b81cb3445b9e19b8c51e97cbe7e9"},"schema_version":"1.0","source":{"id":"1801.08733","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1801.08733","created_at":"2026-05-18T00:25:03Z"},{"alias_kind":"arxiv_version","alias_value":"1801.08733v1","created_at":"2026-05-18T00:25:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.08733","created_at":"2026-05-18T00:25:03Z"},{"alias_kind":"pith_short_12","alias_value":"AO3QNX27RII2","created_at":"2026-05-18T12:32:13Z"},{"alias_kind":"pith_short_16","alias_value":"AO3QNX27RII2X54H","created_at":"2026-05-18T12:32:13Z"},{"alias_kind":"pith_short_8","alias_value":"AO3QNX27","created_at":"2026-05-18T12:32:13Z"}],"graph_snapshots":[{"event_id":"sha256:a103914905ec02b59385cdcb964c1be8a5d64c2e82751c7d61b653a8becb4239","target":"graph","created_at":"2026-05-18T00:25:03Z","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 say that a set is a multiplicative 3-Sidon set if the equation $s_1s_2s_3=t_1t_2t_3$ does not have a solution consisting of distinct elements taken from this set. In this paper we show that the size of a multiplicative 3-Sidon subset of $\\{1,2,\\dots,n\\}$ is at most $\\pi(n)+\\pi(n/2)+n^{2/3}(\\log n )^{2^{1/3}-1/3+o(1)}$, which improves the previously known best bound $\\pi(n)+\\pi(n/2)+cn^{2/3}\\log n/\\log\\log n$.","authors_text":"P\\'eter P\\'al Pach","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-01-26T09:47:10Z","title":"An improved upper bound for the size of the multiplicative 3-Sidon sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.08733","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":"sha256:8265ae65bfe27a659f9f5d49aa6d811520bfcfe5725b7361ae186fabc50011d5","target":"record","created_at":"2026-05-18T00:25:03Z","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":"bfe7c661bf7bdfaec7ee6b4b26e39cb6f3e221ed72a945c3fe86d1f9d9dc54c2","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-01-26T09:47:10Z","title_canon_sha256":"08d6a89481b65b14c6a3a04a7cd6d7478d04b81cb3445b9e19b8c51e97cbe7e9"},"schema_version":"1.0","source":{"id":"1801.08733","kind":"arxiv","version":1}},"canonical_sha256":"03b706df5f8a11abf787bc2f7bfc999cd0034dd3b71fa23fc3f933a000903dd3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"03b706df5f8a11abf787bc2f7bfc999cd0034dd3b71fa23fc3f933a000903dd3","first_computed_at":"2026-05-18T00:25:03.156023Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:25:03.156023Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NrNc26od2eZOTE5yrV643uP7q8tRYWO/o/CjGnXqeW0BcZK+cCLsSe42ir9KiwmrSlr/qcivle7CYT2G7gv7BQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:25:03.156404Z","signed_message":"canonical_sha256_bytes"},"source_id":"1801.08733","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8265ae65bfe27a659f9f5d49aa6d811520bfcfe5725b7361ae186fabc50011d5","sha256:a103914905ec02b59385cdcb964c1be8a5d64c2e82751c7d61b653a8becb4239"],"state_sha256":"c68cf384c91d09f8a92d36cb4ea2391c0179779cbf6f545cf93f0aba4dce0130"}