{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:3CX65FVT7DLSA2X3E3ZEUPOFPH","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":"d1c375b5bbac34cddc6e533ae0ad219e09ea32574fde381705eb294fc22acf19","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-04-09T18:19:01Z","title_canon_sha256":"cdfb5e09d10b9871e3d5dd17ba1264d5b703d0443d09ef5c74786061ca2ebf96"},"schema_version":"1.0","source":{"id":"1504.02410","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.02410","created_at":"2026-05-18T00:10:24Z"},{"alias_kind":"arxiv_version","alias_value":"1504.02410v3","created_at":"2026-05-18T00:10:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.02410","created_at":"2026-05-18T00:10:24Z"},{"alias_kind":"pith_short_12","alias_value":"3CX65FVT7DLS","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_16","alias_value":"3CX65FVT7DLSA2X3","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_8","alias_value":"3CX65FVT","created_at":"2026-05-18T12:29:02Z"}],"graph_snapshots":[{"event_id":"sha256:80de1351378bbb72572dafea88c6ab5e6fc2870ced54b9f7f99def8e0ec6ffb5","target":"graph","created_at":"2026-05-18T00:10:24Z","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 study sets of the form $A = \\big\\{ n \\in \\mathbb N \\big| \\lVert p(n) \\rVert_{\\mathbb R / \\mathbb Z} \\leq \\varepsilon(n) \\big\\}$ for various real valued polynomials $p$ and decay rates $\\varepsilon$. In particular, we ask when such sets are bases of finite order for the positive integers. We show that generically, $A$ is a basis of order $2$ when $\\operatorname{deg} p \\geq 3$, but not when $\\operatorname{deg} p = 2$, although then $A + A$ still has asymptotic density $1$.","authors_text":"Jakub Konieczny","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-04-09T18:19:01Z","title":"Sets of recurrence as bases for the positive integers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.02410","kind":"arxiv","version":3},"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:a133161633c8fe3bb70635f6144c5a4302652a1cc1cb43df7dec6562a75f9437","target":"record","created_at":"2026-05-18T00:10:24Z","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":"d1c375b5bbac34cddc6e533ae0ad219e09ea32574fde381705eb294fc22acf19","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-04-09T18:19:01Z","title_canon_sha256":"cdfb5e09d10b9871e3d5dd17ba1264d5b703d0443d09ef5c74786061ca2ebf96"},"schema_version":"1.0","source":{"id":"1504.02410","kind":"arxiv","version":3}},"canonical_sha256":"d8afee96b3f8d7206afb26f24a3dc579dc3d1bbece4f65b45c6f704d3a54ed7f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d8afee96b3f8d7206afb26f24a3dc579dc3d1bbece4f65b45c6f704d3a54ed7f","first_computed_at":"2026-05-18T00:10:24.776339Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:10:24.776339Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"6YddkSyt6ru1hlZjt7kfm8pOH3HatZUQVXzpGA1ytkAjWOgc1CJjBQSR9cCgCeP6WaAkqyJOaBv2mjAT32TMCw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:10:24.777060Z","signed_message":"canonical_sha256_bytes"},"source_id":"1504.02410","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a133161633c8fe3bb70635f6144c5a4302652a1cc1cb43df7dec6562a75f9437","sha256:80de1351378bbb72572dafea88c6ab5e6fc2870ced54b9f7f99def8e0ec6ffb5"],"state_sha256":"9c1dbbb7dbe569c86b16a319144f26f3799743ba9b2df11c221ce2ada93c0974"}