{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2007:B6RPDKQ7DJR67TJLVG36MFTGTP","short_pith_number":"pith:B6RPDKQ7","schema_version":"1.0","canonical_sha256":"0fa2f1aa1f1a63efcd2ba9b7e616669bf65841c80026664a6439be122966335d","source":{"kind":"arxiv","id":"0711.0185","version":1},"attestation_state":"computed","paper":{"title":"The true complexity of a system of linear equations","license":"","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"J. Wolf (University of Cambridge), W.T. Gowers","submitted_at":"2007-11-01T18:56:56Z","abstract_excerpt":"It is well-known that if a subset A of a finite Abelian group G satisfies a quasirandomness property called uniformity of degree k, then it contains roughly the expected number of arithmetic progressions of length k, that is, the number of progressions one would expect in a random subset of G of the same density as A. One is naturally led to ask which degree of uniformity is required of A in order to control the number of solutions to a general system of linear equations. Using so-called \"quadratic Fourier analysis\", we show that certain linear systems that were previously thought to require q"},"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":"0711.0185","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.NT","submitted_at":"2007-11-01T18:56:56Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"66ec7f239d51792b6a0c3d8ec9fee927d3f8320c60dd799783181efa5ead69cc","abstract_canon_sha256":"9ed03e1187f7a89230818b610ce2b2f2d1ea1efd319ff98f9e6b6dd1f53441e4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:58:12.128348Z","signature_b64":"czQoljcXEtKQT2YwZCogu49KqrLPZSYm/OmdyqGKktee5bjCXZHA1VpYLNhg6pjiCFdQWB8CrcGpdtLWJSNcBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0fa2f1aa1f1a63efcd2ba9b7e616669bf65841c80026664a6439be122966335d","last_reissued_at":"2026-05-18T02:58:12.127663Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:58:12.127663Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The true complexity of a system of linear equations","license":"","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"J. Wolf (University of Cambridge), W.T. Gowers","submitted_at":"2007-11-01T18:56:56Z","abstract_excerpt":"It is well-known that if a subset A of a finite Abelian group G satisfies a quasirandomness property called uniformity of degree k, then it contains roughly the expected number of arithmetic progressions of length k, that is, the number of progressions one would expect in a random subset of G of the same density as A. One is naturally led to ask which degree of uniformity is required of A in order to control the number of solutions to a general system of linear equations. Using so-called \"quadratic Fourier analysis\", we show that certain linear systems that were previously thought to require q"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0711.0185","kind":"arxiv","version":1},"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":"0711.0185","created_at":"2026-05-18T02:58:12.127772+00:00"},{"alias_kind":"arxiv_version","alias_value":"0711.0185v1","created_at":"2026-05-18T02:58:12.127772+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0711.0185","created_at":"2026-05-18T02:58:12.127772+00:00"},{"alias_kind":"pith_short_12","alias_value":"B6RPDKQ7DJR6","created_at":"2026-05-18T12:25:55.427421+00:00"},{"alias_kind":"pith_short_16","alias_value":"B6RPDKQ7DJR67TJL","created_at":"2026-05-18T12:25:55.427421+00:00"},{"alias_kind":"pith_short_8","alias_value":"B6RPDKQ7","created_at":"2026-05-18T12:25:55.427421+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/B6RPDKQ7DJR67TJLVG36MFTGTP","json":"https://pith.science/pith/B6RPDKQ7DJR67TJLVG36MFTGTP.json","graph_json":"https://pith.science/api/pith-number/B6RPDKQ7DJR67TJLVG36MFTGTP/graph.json","events_json":"https://pith.science/api/pith-number/B6RPDKQ7DJR67TJLVG36MFTGTP/events.json","paper":"https://pith.science/paper/B6RPDKQ7"},"agent_actions":{"view_html":"https://pith.science/pith/B6RPDKQ7DJR67TJLVG36MFTGTP","download_json":"https://pith.science/pith/B6RPDKQ7DJR67TJLVG36MFTGTP.json","view_paper":"https://pith.science/paper/B6RPDKQ7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0711.0185&json=true","fetch_graph":"https://pith.science/api/pith-number/B6RPDKQ7DJR67TJLVG36MFTGTP/graph.json","fetch_events":"https://pith.science/api/pith-number/B6RPDKQ7DJR67TJLVG36MFTGTP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/B6RPDKQ7DJR67TJLVG36MFTGTP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/B6RPDKQ7DJR67TJLVG36MFTGTP/action/storage_attestation","attest_author":"https://pith.science/pith/B6RPDKQ7DJR67TJLVG36MFTGTP/action/author_attestation","sign_citation":"https://pith.science/pith/B6RPDKQ7DJR67TJLVG36MFTGTP/action/citation_signature","submit_replication":"https://pith.science/pith/B6RPDKQ7DJR67TJLVG36MFTGTP/action/replication_record"}},"created_at":"2026-05-18T02:58:12.127772+00:00","updated_at":"2026-05-18T02:58:12.127772+00:00"}