{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:TAMNJ3IZELKHY5BPD3AHRTK5VZ","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":"934c45dcac8b9f8618a99033b2d5da6bced0275fc75667e75fcedd132fe9af44","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-01-16T21:54:30Z","title_canon_sha256":"c776101ea5529c8522bc077fd0834b976dedcd42120b2f0900b4ec8a8617c7c0"},"schema_version":"1.0","source":{"id":"1501.04121","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.04121","created_at":"2026-05-18T00:40:08Z"},{"alias_kind":"arxiv_version","alias_value":"1501.04121v1","created_at":"2026-05-18T00:40:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.04121","created_at":"2026-05-18T00:40:08Z"},{"alias_kind":"pith_short_12","alias_value":"TAMNJ3IZELKH","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_16","alias_value":"TAMNJ3IZELKHY5BP","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_8","alias_value":"TAMNJ3IZ","created_at":"2026-05-18T12:29:42Z"}],"graph_snapshots":[{"event_id":"sha256:a8d4cc3cb01467c761228c52406086cafae6278ba5fad0df9642c75f300d4f54","target":"graph","created_at":"2026-05-18T00:40:08Z","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":"Various authors, including McNew, Nathanson and O'Bryant, have recently studied the maximal asymptotic density of a geometric progression free sequence of positive integers. In this paper we prove the existence of geometric progression free sequences with small gaps, partially answering a question posed originally by Beiglb\\\"ock et al. Using probabilistic methods we prove the existence of a sequence $T$ not containing any $6$-term geometric progressions such that for any $x\\geq1$ and $\\varepsilon>0$ the interval $[x,x+C_{\\varepsilon}\\exp((C+\\varepsilon)\\log x/\\log\\log x)]$ contains an element ","authors_text":"Xiaoyu He","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-01-16T21:54:30Z","title":"Geometric Progression-Free Sequences with Small Gaps"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.04121","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:c5438b280291d589befe29d82b3b29364e5ad0ef476804d17454245cff137459","target":"record","created_at":"2026-05-18T00:40:08Z","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":"934c45dcac8b9f8618a99033b2d5da6bced0275fc75667e75fcedd132fe9af44","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-01-16T21:54:30Z","title_canon_sha256":"c776101ea5529c8522bc077fd0834b976dedcd42120b2f0900b4ec8a8617c7c0"},"schema_version":"1.0","source":{"id":"1501.04121","kind":"arxiv","version":1}},"canonical_sha256":"9818d4ed1922d47c742f1ec078cd5dae5bc0c47232c32956b41281f2cd2b860d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9818d4ed1922d47c742f1ec078cd5dae5bc0c47232c32956b41281f2cd2b860d","first_computed_at":"2026-05-18T00:40:08.660726Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:40:08.660726Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"PHsMs9BxI6G1eWYXaQEJNz6HaKrKDfDNOda6MAI6gjRSGP79Nf4OfxuAmQ/W8Afoj8tu0j1RUYlvWceRjn7CAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:40:08.661262Z","signed_message":"canonical_sha256_bytes"},"source_id":"1501.04121","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c5438b280291d589befe29d82b3b29364e5ad0ef476804d17454245cff137459","sha256:a8d4cc3cb01467c761228c52406086cafae6278ba5fad0df9642c75f300d4f54"],"state_sha256":"0d1918e164a1758fe5bf86fc81cc2d885746a3d3a33ff452cba728596126cd24"}