{"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"}