{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:PLVGOU54CVGJLGMU63QBQ3JYAH","short_pith_number":"pith:PLVGOU54","canonical_record":{"source":{"id":"1703.00609","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-03-02T04:18:43Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"f189945ee32467ffa19cd2882c1eae2ad3589803307834770e95990f4a092205","abstract_canon_sha256":"0e294d81663fbec1c9cfdb3dca1d2250ff0fcc385164a0caa152f333b15bfc55"},"schema_version":"1.0"},"canonical_sha256":"7aea6753bc154c959994f6e0186d3801d3b808d5a8df965d5a61d62206da9f3d","source":{"kind":"arxiv","id":"1703.00609","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1703.00609","created_at":"2026-05-18T00:40:13Z"},{"alias_kind":"arxiv_version","alias_value":"1703.00609v2","created_at":"2026-05-18T00:40:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.00609","created_at":"2026-05-18T00:40:13Z"},{"alias_kind":"pith_short_12","alias_value":"PLVGOU54CVGJ","created_at":"2026-05-18T12:31:37Z"},{"alias_kind":"pith_short_16","alias_value":"PLVGOU54CVGJLGMU","created_at":"2026-05-18T12:31:37Z"},{"alias_kind":"pith_short_8","alias_value":"PLVGOU54","created_at":"2026-05-18T12:31:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:PLVGOU54CVGJLGMU63QBQ3JYAH","target":"record","payload":{"canonical_record":{"source":{"id":"1703.00609","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-03-02T04:18:43Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"f189945ee32467ffa19cd2882c1eae2ad3589803307834770e95990f4a092205","abstract_canon_sha256":"0e294d81663fbec1c9cfdb3dca1d2250ff0fcc385164a0caa152f333b15bfc55"},"schema_version":"1.0"},"canonical_sha256":"7aea6753bc154c959994f6e0186d3801d3b808d5a8df965d5a61d62206da9f3d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:40:13.275533Z","signature_b64":"LUuwbuEv2HUWvDxA1GT6ZwMZViLKs/vbVFm3eriIlOFWvZdkQTtsOkSb2pq0hr+q0vrFTIXhD3wbwmhWb3L8BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7aea6753bc154c959994f6e0186d3801d3b808d5a8df965d5a61d62206da9f3d","last_reissued_at":"2026-05-18T00:40:13.274839Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:40:13.274839Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1703.00609","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":"2026-05-18T00:40:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"p5l0VfRCV3NZbDY4jsusbGoYGbbzGRsKQaXfugDYHk+35ZpqauLHrndcBhuhXHE+e2ZX5vRhuXsJ6/4fYYu/Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-02T23:28:47.668096Z"},"content_sha256":"7547bfafd3bf3b0d592b84a915b6e447fdf849aa9be4264e4acf064cc0c27d59","schema_version":"1.0","event_id":"sha256:7547bfafd3bf3b0d592b84a915b6e447fdf849aa9be4264e4acf064cc0c27d59"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:PLVGOU54CVGJLGMU63QBQ3JYAH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The generalized k-resultant modulus set problem in finite fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.CO","authors_text":"David Covert, Doowon Koh, Youngjin Pi","submitted_at":"2017-03-02T04:18:43Z","abstract_excerpt":"Let $\\mathbb F_q^d$ be the $d$-dimensional vector space over the finite field $\\mathbb F_q$ with $q$ elements. Given $k$ sets $E_j\\subset \\mathbb F_q^d$ for $j=1,2,\\ldots, k$, the generalized $k$-resultant modulus set, denoted by $\\Delta_k(E_1,E_2, \\ldots, E_k)$, is defined by $$ \\Delta_k(E_1,E_2, \\ldots, E_k)=\\left\\{\\|{\\bf x}^1+{\\bf x}^2+\\cdots+{\\bf x}^k\\|\\in \\mathbb F_q:{\\bf x}^j\\in E_j,\\, j=1,2,\\ldots, k\\right\\},$$ where $\\|{\\bf y}\\|={\\bf y}_1^2+ \\cdots + {\\bf y}_d^2$ for ${\\bf y}=({\\bf y}_1, \\ldots, {\\bf y}_d)\\in \\mathbb F_q^d.$ We prove that if $\\prod\\limits_{j=1}^3 |E_j| \\ge C q^{3\\left("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.00609","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":"2026-05-18T00:40:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YHMxQqrG0w+C9oRUrGWu9v+w916NILunLohOdjFotJMeHIEagQhTJrasqDUToCfHWPH0UO8ogMpR8yd617SABA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-02T23:28:47.668467Z"},"content_sha256":"16d6bc37c5e7dffe64eca57ebbe586b167d5d5b8cf0e042eeb3e119ad84765b6","schema_version":"1.0","event_id":"sha256:16d6bc37c5e7dffe64eca57ebbe586b167d5d5b8cf0e042eeb3e119ad84765b6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PLVGOU54CVGJLGMU63QBQ3JYAH/bundle.json","state_url":"https://pith.science/pith/PLVGOU54CVGJLGMU63QBQ3JYAH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PLVGOU54CVGJLGMU63QBQ3JYAH/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":"2026-07-02T23:28:47Z","links":{"resolver":"https://pith.science/pith/PLVGOU54CVGJLGMU63QBQ3JYAH","bundle":"https://pith.science/pith/PLVGOU54CVGJLGMU63QBQ3JYAH/bundle.json","state":"https://pith.science/pith/PLVGOU54CVGJLGMU63QBQ3JYAH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PLVGOU54CVGJLGMU63QBQ3JYAH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:PLVGOU54CVGJLGMU63QBQ3JYAH","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":"0e294d81663fbec1c9cfdb3dca1d2250ff0fcc385164a0caa152f333b15bfc55","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-03-02T04:18:43Z","title_canon_sha256":"f189945ee32467ffa19cd2882c1eae2ad3589803307834770e95990f4a092205"},"schema_version":"1.0","source":{"id":"1703.00609","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1703.00609","created_at":"2026-05-18T00:40:13Z"},{"alias_kind":"arxiv_version","alias_value":"1703.00609v2","created_at":"2026-05-18T00:40:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.00609","created_at":"2026-05-18T00:40:13Z"},{"alias_kind":"pith_short_12","alias_value":"PLVGOU54CVGJ","created_at":"2026-05-18T12:31:37Z"},{"alias_kind":"pith_short_16","alias_value":"PLVGOU54CVGJLGMU","created_at":"2026-05-18T12:31:37Z"},{"alias_kind":"pith_short_8","alias_value":"PLVGOU54","created_at":"2026-05-18T12:31:37Z"}],"graph_snapshots":[{"event_id":"sha256:16d6bc37c5e7dffe64eca57ebbe586b167d5d5b8cf0e042eeb3e119ad84765b6","target":"graph","created_at":"2026-05-18T00:40:13Z","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":"Let $\\mathbb F_q^d$ be the $d$-dimensional vector space over the finite field $\\mathbb F_q$ with $q$ elements. Given $k$ sets $E_j\\subset \\mathbb F_q^d$ for $j=1,2,\\ldots, k$, the generalized $k$-resultant modulus set, denoted by $\\Delta_k(E_1,E_2, \\ldots, E_k)$, is defined by $$ \\Delta_k(E_1,E_2, \\ldots, E_k)=\\left\\{\\|{\\bf x}^1+{\\bf x}^2+\\cdots+{\\bf x}^k\\|\\in \\mathbb F_q:{\\bf x}^j\\in E_j,\\, j=1,2,\\ldots, k\\right\\},$$ where $\\|{\\bf y}\\|={\\bf y}_1^2+ \\cdots + {\\bf y}_d^2$ for ${\\bf y}=({\\bf y}_1, \\ldots, {\\bf y}_d)\\in \\mathbb F_q^d.$ We prove that if $\\prod\\limits_{j=1}^3 |E_j| \\ge C q^{3\\left(","authors_text":"David Covert, Doowon Koh, Youngjin Pi","cross_cats":["math.CA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-03-02T04:18:43Z","title":"The generalized k-resultant modulus set problem in finite fields"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.00609","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":"sha256:7547bfafd3bf3b0d592b84a915b6e447fdf849aa9be4264e4acf064cc0c27d59","target":"record","created_at":"2026-05-18T00:40:13Z","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":"0e294d81663fbec1c9cfdb3dca1d2250ff0fcc385164a0caa152f333b15bfc55","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-03-02T04:18:43Z","title_canon_sha256":"f189945ee32467ffa19cd2882c1eae2ad3589803307834770e95990f4a092205"},"schema_version":"1.0","source":{"id":"1703.00609","kind":"arxiv","version":2}},"canonical_sha256":"7aea6753bc154c959994f6e0186d3801d3b808d5a8df965d5a61d62206da9f3d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7aea6753bc154c959994f6e0186d3801d3b808d5a8df965d5a61d62206da9f3d","first_computed_at":"2026-05-18T00:40:13.274839Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:40:13.274839Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LUuwbuEv2HUWvDxA1GT6ZwMZViLKs/vbVFm3eriIlOFWvZdkQTtsOkSb2pq0hr+q0vrFTIXhD3wbwmhWb3L8BQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:40:13.275533Z","signed_message":"canonical_sha256_bytes"},"source_id":"1703.00609","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7547bfafd3bf3b0d592b84a915b6e447fdf849aa9be4264e4acf064cc0c27d59","sha256:16d6bc37c5e7dffe64eca57ebbe586b167d5d5b8cf0e042eeb3e119ad84765b6"],"state_sha256":"b68be539229879c295764227cafb3c04390d11c20b368d2e5304f05b39d36a19"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5VokeOeTBgjLGN0Yv+WlueUDY63GYbz6kg+vcfDIVVs9pqsPMSbUt6VLcNcoYZjBogHX1i43+A/97M4cmJ0BAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-02T23:28:47.670545Z","bundle_sha256":"927bffad67735c869fcbb5adfd9858df980c5a63cad1e3cf417b2dbaa63f1c0c"}}