{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:FHVC6N44AJBR2ZJ3EMWFNBKBOJ","short_pith_number":"pith:FHVC6N44","schema_version":"1.0","canonical_sha256":"29ea2f379c02431d653b232c568541726272ae1b4766ce1f8d58f9688e3ab112","source":{"kind":"arxiv","id":"1305.0043","version":2},"attestation_state":"computed","paper":{"title":"Roth's Theorem in the Piatetski-Shapiro primes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Mariusz Mirek","submitted_at":"2013-04-30T22:30:36Z","abstract_excerpt":"Let $\\mathbf{P}$ denote the set of prime numbers and, for an appropriate function $h$, define a set $\\mathbf{P}_{h}=\\{p\\in\\mathbf{P}: \\exists_{n\\in\\mathbb{N}}\\ p=\\lfloor h(n)\\rfloor\\}$. The aim of this paper is to show that every subset of $\\mathbf{P}_{h}$ having positive relative upper density contains a nontrivial three-term arithmetic progression. In particular the set of Piatetski--Shapiro primes of fixed type $71/72<\\gamma<1$, i.e. $\\{p\\in\\mathbf{P}: \\exists_{n\\in\\mathbb{N}}\\ p=\\lfloor n^{1/\\gamma}\\rfloor\\}$ has this feature. We show this by proving the counterpart of Bourgain--Green's re"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1305.0043","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-04-30T22:30:36Z","cross_cats_sorted":[],"title_canon_sha256":"54a5d10cd0a0a1f93e73f8c480706dd3103e53aacdda4ed5a9fbbc2f79b9e51f","abstract_canon_sha256":"b9dcb161672086921ccad43c77cd687d0205003fbf4eaefb75e2e52481a74b33"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:54:30.796898Z","signature_b64":"xJc82Ek/C2MlaVR8athFtdxAvFlp+pXh0cgSh9UB7AZYOmhn9fayMRRi9VcFg2gZl1Go+BsqBV3JS2YdCfQlAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"29ea2f379c02431d653b232c568541726272ae1b4766ce1f8d58f9688e3ab112","last_reissued_at":"2026-05-18T02:54:30.796442Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:54:30.796442Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Roth's Theorem in the Piatetski-Shapiro primes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Mariusz Mirek","submitted_at":"2013-04-30T22:30:36Z","abstract_excerpt":"Let $\\mathbf{P}$ denote the set of prime numbers and, for an appropriate function $h$, define a set $\\mathbf{P}_{h}=\\{p\\in\\mathbf{P}: \\exists_{n\\in\\mathbb{N}}\\ p=\\lfloor h(n)\\rfloor\\}$. The aim of this paper is to show that every subset of $\\mathbf{P}_{h}$ having positive relative upper density contains a nontrivial three-term arithmetic progression. In particular the set of Piatetski--Shapiro primes of fixed type $71/72<\\gamma<1$, i.e. $\\{p\\in\\mathbf{P}: \\exists_{n\\in\\mathbb{N}}\\ p=\\lfloor n^{1/\\gamma}\\rfloor\\}$ has this feature. We show this by proving the counterpart of Bourgain--Green's re"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.0043","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1305.0043","created_at":"2026-05-18T02:54:30.796512+00:00"},{"alias_kind":"arxiv_version","alias_value":"1305.0043v2","created_at":"2026-05-18T02:54:30.796512+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.0043","created_at":"2026-05-18T02:54:30.796512+00:00"},{"alias_kind":"pith_short_12","alias_value":"FHVC6N44AJBR","created_at":"2026-05-18T12:27:45.050594+00:00"},{"alias_kind":"pith_short_16","alias_value":"FHVC6N44AJBR2ZJ3","created_at":"2026-05-18T12:27:45.050594+00:00"},{"alias_kind":"pith_short_8","alias_value":"FHVC6N44","created_at":"2026-05-18T12:27:45.050594+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/FHVC6N44AJBR2ZJ3EMWFNBKBOJ","json":"https://pith.science/pith/FHVC6N44AJBR2ZJ3EMWFNBKBOJ.json","graph_json":"https://pith.science/api/pith-number/FHVC6N44AJBR2ZJ3EMWFNBKBOJ/graph.json","events_json":"https://pith.science/api/pith-number/FHVC6N44AJBR2ZJ3EMWFNBKBOJ/events.json","paper":"https://pith.science/paper/FHVC6N44"},"agent_actions":{"view_html":"https://pith.science/pith/FHVC6N44AJBR2ZJ3EMWFNBKBOJ","download_json":"https://pith.science/pith/FHVC6N44AJBR2ZJ3EMWFNBKBOJ.json","view_paper":"https://pith.science/paper/FHVC6N44","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1305.0043&json=true","fetch_graph":"https://pith.science/api/pith-number/FHVC6N44AJBR2ZJ3EMWFNBKBOJ/graph.json","fetch_events":"https://pith.science/api/pith-number/FHVC6N44AJBR2ZJ3EMWFNBKBOJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FHVC6N44AJBR2ZJ3EMWFNBKBOJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FHVC6N44AJBR2ZJ3EMWFNBKBOJ/action/storage_attestation","attest_author":"https://pith.science/pith/FHVC6N44AJBR2ZJ3EMWFNBKBOJ/action/author_attestation","sign_citation":"https://pith.science/pith/FHVC6N44AJBR2ZJ3EMWFNBKBOJ/action/citation_signature","submit_replication":"https://pith.science/pith/FHVC6N44AJBR2ZJ3EMWFNBKBOJ/action/replication_record"}},"created_at":"2026-05-18T02:54:30.796512+00:00","updated_at":"2026-05-18T02:54:30.796512+00:00"}