{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2006:TFCDBMM6EUD3ARJVYXTJSFZQXC","short_pith_number":"pith:TFCDBMM6","canonical_record":{"source":{"id":"math/0607692","kind":"arxiv","version":3},"metadata":{"license":"","primary_cat":"math.NT","submitted_at":"2006-07-27T00:13:43Z","cross_cats_sorted":[],"title_canon_sha256":"64fbaf4fe87200eb72537acb857f49f3946bc9479090ade8a9f4d4b00ad0eb79","abstract_canon_sha256":"3abca24f394abde4aa70e6bde5024ede0248d04a3dff4a2afbbe0e9f48465264"},"schema_version":"1.0"},"canonical_sha256":"994430b19e2507b04535c5e6991730b8bda2ce9931d9a3198ce4f5dbab668f23","source":{"kind":"arxiv","id":"math/0607692","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"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2006:TFCDBMM6EUD3ARJVYXTJSFZQXC","target":"record","payload":{"canonical_record":{"source":{"id":"math/0607692","kind":"arxiv","version":3},"metadata":{"license":"","primary_cat":"math.NT","submitted_at":"2006-07-27T00:13:43Z","cross_cats_sorted":[],"title_canon_sha256":"64fbaf4fe87200eb72537acb857f49f3946bc9479090ade8a9f4d4b00ad0eb79","abstract_canon_sha256":"3abca24f394abde4aa70e6bde5024ede0248d04a3dff4a2afbbe0e9f48465264"},"schema_version":"1.0"},"canonical_sha256":"994430b19e2507b04535c5e6991730b8bda2ce9931d9a3198ce4f5dbab668f23","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:57:45.553331Z","signature_b64":"+QJY7O3CxFceNa81sG+zrxp560o3MeZ0FrQWiWiy3pZ4mLUlQueNlW0CfaBuqQVJ093ghDxoMAW5EB66KOJdDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"994430b19e2507b04535c5e6991730b8bda2ce9931d9a3198ce4f5dbab668f23","last_reissued_at":"2026-05-18T02:57:45.552746Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:57:45.552746Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0607692","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:57:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6QQKe2m2dn7zjgnnr7egKf8LRczN5B20/LcI94TZjrXNyaKQQjVx6ZVs38qP87gbDw7y3+OKTxIBRdAL9DNUCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T03:04:42.363208Z"},"content_sha256":"a37437e710b060d95b01f6fd57db3389f6be6cce3cf7420ebb3165a8e0ed58ba","schema_version":"1.0","event_id":"sha256:a37437e710b060d95b01f6fd57db3389f6be6cce3cf7420ebb3165a8e0ed58ba"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2006:TFCDBMM6EUD3ARJVYXTJSFZQXC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Density of non-residues in Burgess-type intervals and applications","license":"","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"D. R. Heath-Brown, I. E. Shparlinski, M. Z. Garaev, W. D. Banks","submitted_at":"2006-07-27T00:13:43Z","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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0607692","kind":"arxiv","version":3},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:57:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8Y/xspmxour54WsU2vGTad2GT6xK1N5b2+g/YfB3rJwD7SgQGUkmMvj5+87OpADTfwgjGir1vEFnhHGJvoV0Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T03:04:42.363565Z"},"content_sha256":"42cc9dad3af95102347918691c46db24f4b9d04ad5a4457aa5cdb63455ad486d","schema_version":"1.0","event_id":"sha256:42cc9dad3af95102347918691c46db24f4b9d04ad5a4457aa5cdb63455ad486d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TFCDBMM6EUD3ARJVYXTJSFZQXC/bundle.json","state_url":"https://pith.science/pith/TFCDBMM6EUD3ARJVYXTJSFZQXC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TFCDBMM6EUD3ARJVYXTJSFZQXC/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-04T03:04:42Z","links":{"resolver":"https://pith.science/pith/TFCDBMM6EUD3ARJVYXTJSFZQXC","bundle":"https://pith.science/pith/TFCDBMM6EUD3ARJVYXTJSFZQXC/bundle.json","state":"https://pith.science/pith/TFCDBMM6EUD3ARJVYXTJSFZQXC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TFCDBMM6EUD3ARJVYXTJSFZQXC/bundle.json"},"state":{"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"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AH6yWnnxEVYYDJKKUu8bLn1QLkomrfhXFfpA+6n/7sLGCO2ecswiotCOZ7VNN0LzoAAex0odzM2q0HIBsr3sDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T03:04:42.365764Z","bundle_sha256":"ff447a378b7514de8811337da6a5f3bd541b56642c3081ad3edd48e25024b63c"}}