{"paper":{"title":"Distributions of weights and a question of Wilf","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Michael Hellus, Rolf Waldi","submitted_at":"2018-04-17T10:07:11Z","abstract_excerpt":"Let $S$ be a numerical semigroup of embedding dimension $e$ and conductor $c$. The question of Wilf is, if $\\#(\\mathbb N\\setminus S)/c\\leq e-1/e$.\n  \\noindent In (An asymptotic result concerning a question of Wilf, arXiv:1111.2779v1 [math.CO], 2011, Lemma 3), Zhai has shown an analogous inequality for the distribution of weights $x\\cdot\\gamma$, $x\\in\\mathbb N^d$, w.\\,r. to a positive weight vector $\\gamma$:\n  \\noindent Let $B\\subseteq\\mathbb N^d$ be finite and the complement of an $\\mathbb N^d$-ideal. Denote by $\\operatorname{mean}(B\\cdot\\gamma)$ the average weight of $B$. Then \\[\\operatorname"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.06146","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"}