{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:TFCDBMM6EUD3ARJVYXTJSFZQXC","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":"3abca24f394abde4aa70e6bde5024ede0248d04a3dff4a2afbbe0e9f48465264","cross_cats_sorted":[],"license":"","primary_cat":"math.NT","submitted_at":"2006-07-27T00:13:43Z","title_canon_sha256":"64fbaf4fe87200eb72537acb857f49f3946bc9479090ade8a9f4d4b00ad0eb79"},"schema_version":"1.0","source":{"id":"math/0607692","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0607692","created_at":"2026-05-18T02:57:45Z"},{"alias_kind":"arxiv_version","alias_value":"math/0607692v3","created_at":"2026-05-18T02:57:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0607692","created_at":"2026-05-18T02:57:45Z"},{"alias_kind":"pith_short_12","alias_value":"TFCDBMM6EUD3","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_16","alias_value":"TFCDBMM6EUD3ARJV","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_8","alias_value":"TFCDBMM6","created_at":"2026-05-18T12:25:54Z"}],"graph_snapshots":[{"event_id":"sha256:42cc9dad3af95102347918691c46db24f4b9d04ad5a4457aa5cdb63455ad486d","target":"graph","created_at":"2026-05-18T02:57:45Z","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 show that for any fixed $\\eps>0$, there are numbers $\\delta>0$ and $p_0\\ge 2$ with the following property: for every prime $p\\ge p_0$ and every integer $N$ such that $p^{1/(4\\sqrt{e})+\\eps}\\le N\\le p$, the sequence $1,2,...,N$ contains at least $\\delta N$ quadratic non-residues modulo $p$. We use this result to obtain strong upper bounds on the sizes of the least quadratic non-residues in Beatty and Piatetski--Shapiro sequences.","authors_text":"D. R. Heath-Brown, I. E. Shparlinski, M. Z. Garaev, W. D. Banks","cross_cats":[],"headline":"","license":"","primary_cat":"math.NT","submitted_at":"2006-07-27T00:13:43Z","title":"Density of non-residues in Burgess-type intervals and applications"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0607692","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:a37437e710b060d95b01f6fd57db3389f6be6cce3cf7420ebb3165a8e0ed58ba","target":"record","created_at":"2026-05-18T02:57:45Z","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":"3abca24f394abde4aa70e6bde5024ede0248d04a3dff4a2afbbe0e9f48465264","cross_cats_sorted":[],"license":"","primary_cat":"math.NT","submitted_at":"2006-07-27T00:13:43Z","title_canon_sha256":"64fbaf4fe87200eb72537acb857f49f3946bc9479090ade8a9f4d4b00ad0eb79"},"schema_version":"1.0","source":{"id":"math/0607692","kind":"arxiv","version":3}},"canonical_sha256":"994430b19e2507b04535c5e6991730b8bda2ce9931d9a3198ce4f5dbab668f23","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"994430b19e2507b04535c5e6991730b8bda2ce9931d9a3198ce4f5dbab668f23","first_computed_at":"2026-05-18T02:57:45.552746Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:57:45.552746Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+QJY7O3CxFceNa81sG+zrxp560o3MeZ0FrQWiWiy3pZ4mLUlQueNlW0CfaBuqQVJ093ghDxoMAW5EB66KOJdDw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:57:45.553331Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0607692","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a37437e710b060d95b01f6fd57db3389f6be6cce3cf7420ebb3165a8e0ed58ba","sha256:42cc9dad3af95102347918691c46db24f4b9d04ad5a4457aa5cdb63455ad486d"],"state_sha256":"4b8311ae73489f248c3ec20faba3fa03add1aa75e0e5559f2d0172759e8092e9"}