{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:7KRSTS5AXDFWB7AVPSVGYQHCC5","short_pith_number":"pith:7KRSTS5A","canonical_record":{"source":{"id":"1108.5832","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-08-30T05:33:20Z","cross_cats_sorted":["math.CO","math.CV"],"title_canon_sha256":"ad2989751b7dfbb85758459cee7fc0c3fb2f407051869d8e19163c554ceff847","abstract_canon_sha256":"4505428865c339b4fbff7f38973ccde6d260f8c9481576c2280796095f339a03"},"schema_version":"1.0"},"canonical_sha256":"faa329cba0b8cb60fc157caa6c40e21742efdd50d763f82a1f7dba5dd04db560","source":{"kind":"arxiv","id":"1108.5832","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1108.5832","created_at":"2026-05-18T04:14:28Z"},{"alias_kind":"arxiv_version","alias_value":"1108.5832v1","created_at":"2026-05-18T04:14:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.5832","created_at":"2026-05-18T04:14:28Z"},{"alias_kind":"pith_short_12","alias_value":"7KRSTS5AXDFW","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"7KRSTS5AXDFWB7AV","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"7KRSTS5A","created_at":"2026-05-18T12:26:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:7KRSTS5AXDFWB7AVPSVGYQHCC5","target":"record","payload":{"canonical_record":{"source":{"id":"1108.5832","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-08-30T05:33:20Z","cross_cats_sorted":["math.CO","math.CV"],"title_canon_sha256":"ad2989751b7dfbb85758459cee7fc0c3fb2f407051869d8e19163c554ceff847","abstract_canon_sha256":"4505428865c339b4fbff7f38973ccde6d260f8c9481576c2280796095f339a03"},"schema_version":"1.0"},"canonical_sha256":"faa329cba0b8cb60fc157caa6c40e21742efdd50d763f82a1f7dba5dd04db560","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:14:28.468089Z","signature_b64":"r7+6Ij7n/jW4LLzOqPwtVOU4/0MFalhO2UJPrRDV091ax88HGEaEwi9uzIwk8FcOBI6GLwUFJjuHYgCk6WjnDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"faa329cba0b8cb60fc157caa6c40e21742efdd50d763f82a1f7dba5dd04db560","last_reissued_at":"2026-05-18T04:14:28.467551Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:14:28.467551Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1108.5832","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:14:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YXUlRr9/3a3NLYMywZD1OoBZFC9L1+Y5Pgf1dmfaRnNtAnwAOo5mAy+2mBxxt4ap98JA1cXwuRTPCQDtgApHBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T09:53:13.695084Z"},"content_sha256":"2dbda39e08190aeb48a7edb4c688f47b0947491fdb3066f4c2b2fd5743ae0870","schema_version":"1.0","event_id":"sha256:2dbda39e08190aeb48a7edb4c688f47b0947491fdb3066f4c2b2fd5743ae0870"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:7KRSTS5AXDFWB7AVPSVGYQHCC5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A question of S\\'{a}rkozy and S\\'{o}s on representation functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.CV"],"primary_cat":"math.NT","authors_text":"Lianrong Ma, Yan Li","submitted_at":"2011-08-30T05:33:20Z","abstract_excerpt":"For $m\\geq 1$, let $0<b_0<b_1<...<b_m$ and $\\ e_0,e_1,...,e_m>0$ be fixed positive integers. Assume there exists a prime $p$ and an integer $t>0$ such that $p^t\\mid b_0$, but $p^t\\nmid b_{i}\\ {\\rm for}\\ 1\\leq i\\leq m$. Then, we prove that there is no infinite subset $\\mathcal A$ of positive integers, such that the number of solutions of the following equation $$n=b_0(a_{0,1}+\\cdot +a_{0,e_0})+...+b_m(a_{m,1}+...+a_{m,r_m}),\\ a_{i,j}\\in \\mathcal A$$ is constant for $n$ large enough. This result generalizes the recent result of Cilleruelo and Ru\\'{e} for the bilinear case, and answers a question"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.5832","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:14:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YRJXtczYjvZAsy0mIgvB1UYpqpwiaLk9Sg5UZDXDcnu98zDANag0u/2RfN2yoIeEN4Kliex46ysUHIoJ2YPbDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T09:53:13.695449Z"},"content_sha256":"8b95a183605f172e2a443078b1752dcd232d6580031451589ce4f0b644bbb09c","schema_version":"1.0","event_id":"sha256:8b95a183605f172e2a443078b1752dcd232d6580031451589ce4f0b644bbb09c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7KRSTS5AXDFWB7AVPSVGYQHCC5/bundle.json","state_url":"https://pith.science/pith/7KRSTS5AXDFWB7AVPSVGYQHCC5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7KRSTS5AXDFWB7AVPSVGYQHCC5/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":"2026-06-08T09:53:13Z","links":{"resolver":"https://pith.science/pith/7KRSTS5AXDFWB7AVPSVGYQHCC5","bundle":"https://pith.science/pith/7KRSTS5AXDFWB7AVPSVGYQHCC5/bundle.json","state":"https://pith.science/pith/7KRSTS5AXDFWB7AVPSVGYQHCC5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7KRSTS5AXDFWB7AVPSVGYQHCC5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:7KRSTS5AXDFWB7AVPSVGYQHCC5","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":"4505428865c339b4fbff7f38973ccde6d260f8c9481576c2280796095f339a03","cross_cats_sorted":["math.CO","math.CV"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-08-30T05:33:20Z","title_canon_sha256":"ad2989751b7dfbb85758459cee7fc0c3fb2f407051869d8e19163c554ceff847"},"schema_version":"1.0","source":{"id":"1108.5832","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1108.5832","created_at":"2026-05-18T04:14:28Z"},{"alias_kind":"arxiv_version","alias_value":"1108.5832v1","created_at":"2026-05-18T04:14:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.5832","created_at":"2026-05-18T04:14:28Z"},{"alias_kind":"pith_short_12","alias_value":"7KRSTS5AXDFW","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"7KRSTS5AXDFWB7AV","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"7KRSTS5A","created_at":"2026-05-18T12:26:22Z"}],"graph_snapshots":[{"event_id":"sha256:8b95a183605f172e2a443078b1752dcd232d6580031451589ce4f0b644bbb09c","target":"graph","created_at":"2026-05-18T04:14:28Z","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":"For $m\\geq 1$, let $0<b_0<b_1<...<b_m$ and $\\ e_0,e_1,...,e_m>0$ be fixed positive integers. Assume there exists a prime $p$ and an integer $t>0$ such that $p^t\\mid b_0$, but $p^t\\nmid b_{i}\\ {\\rm for}\\ 1\\leq i\\leq m$. Then, we prove that there is no infinite subset $\\mathcal A$ of positive integers, such that the number of solutions of the following equation $$n=b_0(a_{0,1}+\\cdot +a_{0,e_0})+...+b_m(a_{m,1}+...+a_{m,r_m}),\\ a_{i,j}\\in \\mathcal A$$ is constant for $n$ large enough. This result generalizes the recent result of Cilleruelo and Ru\\'{e} for the bilinear case, and answers a question","authors_text":"Lianrong Ma, Yan Li","cross_cats":["math.CO","math.CV"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-08-30T05:33:20Z","title":"A question of S\\'{a}rkozy and S\\'{o}s on representation functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.5832","kind":"arxiv","version":1},"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":"sha256:2dbda39e08190aeb48a7edb4c688f47b0947491fdb3066f4c2b2fd5743ae0870","target":"record","created_at":"2026-05-18T04:14:28Z","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":"4505428865c339b4fbff7f38973ccde6d260f8c9481576c2280796095f339a03","cross_cats_sorted":["math.CO","math.CV"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-08-30T05:33:20Z","title_canon_sha256":"ad2989751b7dfbb85758459cee7fc0c3fb2f407051869d8e19163c554ceff847"},"schema_version":"1.0","source":{"id":"1108.5832","kind":"arxiv","version":1}},"canonical_sha256":"faa329cba0b8cb60fc157caa6c40e21742efdd50d763f82a1f7dba5dd04db560","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"faa329cba0b8cb60fc157caa6c40e21742efdd50d763f82a1f7dba5dd04db560","first_computed_at":"2026-05-18T04:14:28.467551Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:14:28.467551Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"r7+6Ij7n/jW4LLzOqPwtVOU4/0MFalhO2UJPrRDV091ax88HGEaEwi9uzIwk8FcOBI6GLwUFJjuHYgCk6WjnDw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:14:28.468089Z","signed_message":"canonical_sha256_bytes"},"source_id":"1108.5832","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2dbda39e08190aeb48a7edb4c688f47b0947491fdb3066f4c2b2fd5743ae0870","sha256:8b95a183605f172e2a443078b1752dcd232d6580031451589ce4f0b644bbb09c"],"state_sha256":"f134edd52659bc3e51f906ee18f7228dffe7d4bc3bf08193a6aac634f95dc8a3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"S/pMCcURh9fPKEry8x45/jg7lLEqOM1MipexZWBS52Z3AZTMm6KaqhVXMxDHMLW/FUsqNCZCNiPnHs6Xq6baAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T09:53:13.697460Z","bundle_sha256":"f7a9a1f6e7f878e4016bfad128091f8bd974890a5643d21fc8ecb258b8d14479"}}