{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:AN6VJ6QGJHF6LLDVZAZ6LJFRJW","short_pith_number":"pith:AN6VJ6QG","schema_version":"1.0","canonical_sha256":"037d54fa0649cbe5ac75c833e5a4b14db4f0d81e66871e07099f3d33954aacb7","source":{"kind":"arxiv","id":"1102.5351","version":2},"attestation_state":"computed","paper":{"title":"Arithmetic-Progression-Weighted Subsequence Sums","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Andreas Philipp, David J. Grynkiewicz, Vadim Ponomarenko","submitted_at":"2011-02-25T21:18:20Z","abstract_excerpt":"Let $G$ be an abelian group, let $S$ be a sequence of terms $s_1,s_2,...,s_{n}\\in G$ not all contained in a coset of a proper subgroup of $G$, and let $W$ be a sequence of $n$ consecutive integers. Let $$W\\odot S=\\{w_1s_1+...+w_ns_n:\\;w_i {a term of} W,\\, w_i\\neq w_j{for} i\\neq j\\},$$ which is a particular kind of weighted restricted sumset. We show that $|W\\odot S|\\geq \\min\\{|G|-1,\\,n\\}$, that $W\\odot S=G$ if $n\\geq |G|+1$, and also characterize all sequences $S$ of length $|G|$ with $W\\odot S\\neq G$. This result then allows us to characterize when a linear equation $$a_1x_1+...+a_rx_r\\equiv "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1102.5351","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-02-25T21:18:20Z","cross_cats_sorted":[],"title_canon_sha256":"e02be46be0bb8c361e1fa35c9d7b6561ca40dde00f65317165634a7231616bf7","abstract_canon_sha256":"f9106996b08c61f71bd6ba0e499845c18a1eb1642659ef8264c99fe4f79837f5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:19:13.197470Z","signature_b64":"8lat3WqgEFTE7Ixd0dVhhA3pqZDWejQcZPYXhzqo+3GArbADoBSZjoV+ArkGZr4WjsZ0MmHT96sn3vaOnmPzCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"037d54fa0649cbe5ac75c833e5a4b14db4f0d81e66871e07099f3d33954aacb7","last_reissued_at":"2026-05-18T04:19:13.196978Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:19:13.196978Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Arithmetic-Progression-Weighted Subsequence Sums","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Andreas Philipp, David J. Grynkiewicz, Vadim Ponomarenko","submitted_at":"2011-02-25T21:18:20Z","abstract_excerpt":"Let $G$ be an abelian group, let $S$ be a sequence of terms $s_1,s_2,...,s_{n}\\in G$ not all contained in a coset of a proper subgroup of $G$, and let $W$ be a sequence of $n$ consecutive integers. Let $$W\\odot S=\\{w_1s_1+...+w_ns_n:\\;w_i {a term of} W,\\, w_i\\neq w_j{for} i\\neq j\\},$$ which is a particular kind of weighted restricted sumset. We show that $|W\\odot S|\\geq \\min\\{|G|-1,\\,n\\}$, that $W\\odot S=G$ if $n\\geq |G|+1$, and also characterize all sequences $S$ of length $|G|$ with $W\\odot S\\neq G$. This result then allows us to characterize when a linear equation $$a_1x_1+...+a_rx_r\\equiv "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.5351","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1102.5351","created_at":"2026-05-18T04:19:13.197072+00:00"},{"alias_kind":"arxiv_version","alias_value":"1102.5351v2","created_at":"2026-05-18T04:19:13.197072+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.5351","created_at":"2026-05-18T04:19:13.197072+00:00"},{"alias_kind":"pith_short_12","alias_value":"AN6VJ6QGJHF6","created_at":"2026-05-18T12:26:24.575870+00:00"},{"alias_kind":"pith_short_16","alias_value":"AN6VJ6QGJHF6LLDV","created_at":"2026-05-18T12:26:24.575870+00:00"},{"alias_kind":"pith_short_8","alias_value":"AN6VJ6QG","created_at":"2026-05-18T12:26:24.575870+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/AN6VJ6QGJHF6LLDVZAZ6LJFRJW","json":"https://pith.science/pith/AN6VJ6QGJHF6LLDVZAZ6LJFRJW.json","graph_json":"https://pith.science/api/pith-number/AN6VJ6QGJHF6LLDVZAZ6LJFRJW/graph.json","events_json":"https://pith.science/api/pith-number/AN6VJ6QGJHF6LLDVZAZ6LJFRJW/events.json","paper":"https://pith.science/paper/AN6VJ6QG"},"agent_actions":{"view_html":"https://pith.science/pith/AN6VJ6QGJHF6LLDVZAZ6LJFRJW","download_json":"https://pith.science/pith/AN6VJ6QGJHF6LLDVZAZ6LJFRJW.json","view_paper":"https://pith.science/paper/AN6VJ6QG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1102.5351&json=true","fetch_graph":"https://pith.science/api/pith-number/AN6VJ6QGJHF6LLDVZAZ6LJFRJW/graph.json","fetch_events":"https://pith.science/api/pith-number/AN6VJ6QGJHF6LLDVZAZ6LJFRJW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AN6VJ6QGJHF6LLDVZAZ6LJFRJW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AN6VJ6QGJHF6LLDVZAZ6LJFRJW/action/storage_attestation","attest_author":"https://pith.science/pith/AN6VJ6QGJHF6LLDVZAZ6LJFRJW/action/author_attestation","sign_citation":"https://pith.science/pith/AN6VJ6QGJHF6LLDVZAZ6LJFRJW/action/citation_signature","submit_replication":"https://pith.science/pith/AN6VJ6QGJHF6LLDVZAZ6LJFRJW/action/replication_record"}},"created_at":"2026-05-18T04:19:13.197072+00:00","updated_at":"2026-05-18T04:19:13.197072+00:00"}