{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:CVA5RZ3ULJKHOAAIA2L2QJXDDV","short_pith_number":"pith:CVA5RZ3U","canonical_record":{"source":{"id":"1510.00136","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-10-01T08:10:33Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"70f0df707a8a70b1961b57e2b489d9dcc8a4f6fd629e4955de018b98c878bd18","abstract_canon_sha256":"d150882fe4acb45cdb951cb986f4dcd504070cf5e860a2d73dce3488ca182666"},"schema_version":"1.0"},"canonical_sha256":"1541d8e7745a547700080697a826e31d58c1d7e5e360b93445cf99b1b0002b64","source":{"kind":"arxiv","id":"1510.00136","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.00136","created_at":"2026-05-18T01:24:24Z"},{"alias_kind":"arxiv_version","alias_value":"1510.00136v2","created_at":"2026-05-18T01:24:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.00136","created_at":"2026-05-18T01:24:24Z"},{"alias_kind":"pith_short_12","alias_value":"CVA5RZ3ULJKH","created_at":"2026-05-18T12:29:17Z"},{"alias_kind":"pith_short_16","alias_value":"CVA5RZ3ULJKHOAAI","created_at":"2026-05-18T12:29:17Z"},{"alias_kind":"pith_short_8","alias_value":"CVA5RZ3U","created_at":"2026-05-18T12:29:17Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:CVA5RZ3ULJKHOAAIA2L2QJXDDV","target":"record","payload":{"canonical_record":{"source":{"id":"1510.00136","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-10-01T08:10:33Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"70f0df707a8a70b1961b57e2b489d9dcc8a4f6fd629e4955de018b98c878bd18","abstract_canon_sha256":"d150882fe4acb45cdb951cb986f4dcd504070cf5e860a2d73dce3488ca182666"},"schema_version":"1.0"},"canonical_sha256":"1541d8e7745a547700080697a826e31d58c1d7e5e360b93445cf99b1b0002b64","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:24:24.321181Z","signature_b64":"gP+8GLpiF4w89LVAdOKPmNZrDDM7GpUuweRdYDZi70sXBGn4EJ0BOjs6b5r1zo+EeCkvgbqT8ojzX4lv6vbVAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1541d8e7745a547700080697a826e31d58c1d7e5e360b93445cf99b1b0002b64","last_reissued_at":"2026-05-18T01:24:24.320501Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:24:24.320501Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1510.00136","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:24:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZcOASwKFJLV63Opqtrt7D0ypsCKINC4cMD2pNE4QKZEvPCKyc2y1Z45k5ZTd5rPEkYIJLZm7U2KYv2f6aI93Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T06:51:15.598354Z"},"content_sha256":"4db90b2d020a09521751271068947501b854d81d971428502e889567e4501410","schema_version":"1.0","event_id":"sha256:4db90b2d020a09521751271068947501b854d81d971428502e889567e4501410"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:CVA5RZ3ULJKHOAAIA2L2QJXDDV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A transference approach to a Roth-type theorem in the squares","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Sean Prendiville, Tim Browning","submitted_at":"2015-10-01T08:10:33Z","abstract_excerpt":"We show that any subset of the squares of positive relative upper density contains non-trivial solutions to a translation-invariant linear equation in five or more variables, with explicit quantitative bounds. As a consequence, we establish the partition regularity of any diagonal quadric in five or more variables whose coefficients sum to zero. Unlike previous approaches, which are limited to equations in seven or more variables, we employ transference technology of Green to import bounds from the linear setting."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.00136","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:24:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wzkpGcorm9GZcgeGQrCFdFhSZqmb9r7gU7+f3cRwtJ82Kr+ceM/M5SuPg56CoI1oo0vrTnhvVF9+3UspYWFYBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T06:51:15.598776Z"},"content_sha256":"296e41575b1fe0308e6cbc2694b52fc2d1132e3383f27a09a1e839dd72701d5a","schema_version":"1.0","event_id":"sha256:296e41575b1fe0308e6cbc2694b52fc2d1132e3383f27a09a1e839dd72701d5a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CVA5RZ3ULJKHOAAIA2L2QJXDDV/bundle.json","state_url":"https://pith.science/pith/CVA5RZ3ULJKHOAAIA2L2QJXDDV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CVA5RZ3ULJKHOAAIA2L2QJXDDV/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":"2026-06-25T06:51:15Z","links":{"resolver":"https://pith.science/pith/CVA5RZ3ULJKHOAAIA2L2QJXDDV","bundle":"https://pith.science/pith/CVA5RZ3ULJKHOAAIA2L2QJXDDV/bundle.json","state":"https://pith.science/pith/CVA5RZ3ULJKHOAAIA2L2QJXDDV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CVA5RZ3ULJKHOAAIA2L2QJXDDV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:CVA5RZ3ULJKHOAAIA2L2QJXDDV","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":"d150882fe4acb45cdb951cb986f4dcd504070cf5e860a2d73dce3488ca182666","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-10-01T08:10:33Z","title_canon_sha256":"70f0df707a8a70b1961b57e2b489d9dcc8a4f6fd629e4955de018b98c878bd18"},"schema_version":"1.0","source":{"id":"1510.00136","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.00136","created_at":"2026-05-18T01:24:24Z"},{"alias_kind":"arxiv_version","alias_value":"1510.00136v2","created_at":"2026-05-18T01:24:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.00136","created_at":"2026-05-18T01:24:24Z"},{"alias_kind":"pith_short_12","alias_value":"CVA5RZ3ULJKH","created_at":"2026-05-18T12:29:17Z"},{"alias_kind":"pith_short_16","alias_value":"CVA5RZ3ULJKHOAAI","created_at":"2026-05-18T12:29:17Z"},{"alias_kind":"pith_short_8","alias_value":"CVA5RZ3U","created_at":"2026-05-18T12:29:17Z"}],"graph_snapshots":[{"event_id":"sha256:296e41575b1fe0308e6cbc2694b52fc2d1132e3383f27a09a1e839dd72701d5a","target":"graph","created_at":"2026-05-18T01:24:24Z","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 any subset of the squares of positive relative upper density contains non-trivial solutions to a translation-invariant linear equation in five or more variables, with explicit quantitative bounds. As a consequence, we establish the partition regularity of any diagonal quadric in five or more variables whose coefficients sum to zero. Unlike previous approaches, which are limited to equations in seven or more variables, we employ transference technology of Green to import bounds from the linear setting.","authors_text":"Sean Prendiville, Tim Browning","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-10-01T08:10:33Z","title":"A transference approach to a Roth-type theorem in the squares"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.00136","kind":"arxiv","version":2},"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":"sha256:4db90b2d020a09521751271068947501b854d81d971428502e889567e4501410","target":"record","created_at":"2026-05-18T01:24:24Z","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":"d150882fe4acb45cdb951cb986f4dcd504070cf5e860a2d73dce3488ca182666","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-10-01T08:10:33Z","title_canon_sha256":"70f0df707a8a70b1961b57e2b489d9dcc8a4f6fd629e4955de018b98c878bd18"},"schema_version":"1.0","source":{"id":"1510.00136","kind":"arxiv","version":2}},"canonical_sha256":"1541d8e7745a547700080697a826e31d58c1d7e5e360b93445cf99b1b0002b64","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1541d8e7745a547700080697a826e31d58c1d7e5e360b93445cf99b1b0002b64","first_computed_at":"2026-05-18T01:24:24.320501Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:24:24.320501Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gP+8GLpiF4w89LVAdOKPmNZrDDM7GpUuweRdYDZi70sXBGn4EJ0BOjs6b5r1zo+EeCkvgbqT8ojzX4lv6vbVAw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:24:24.321181Z","signed_message":"canonical_sha256_bytes"},"source_id":"1510.00136","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4db90b2d020a09521751271068947501b854d81d971428502e889567e4501410","sha256:296e41575b1fe0308e6cbc2694b52fc2d1132e3383f27a09a1e839dd72701d5a"],"state_sha256":"b292d1e130581ba21f912e1bedb9d273434968805d8ddad484589d29e322a5ec"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"borrI40rWPTIh3lYcYKdXaUfEawU3nMXZhjTmiyGQJixeFGY7eFCIZ29j0FbH1FczIH5hfT+tO/jYCS6xOZqAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T06:51:15.601020Z","bundle_sha256":"aaad4ee4a9078a9bc83d6776234c794fd10f08e3e9d3281da8fa88268b0f1753"}}