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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"},"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":"1804.06146","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2018-04-17T10:07:11Z","cross_cats_sorted":[],"title_canon_sha256":"8e76a1a08088eaf3322f5a47038c87a2a842128e3a5867b161b3abd81c7bb629","abstract_canon_sha256":"226ea5e5260d6a184a73a3c5a72cc41195a22ade01bf645b2506f10d3c66fd7f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:18:08.774773Z","signature_b64":"HkU1Mi2rUJBXcJZ/G4C09HsmDu3qyDoZFubDv4VpN1r3401GfamkkZnO5mzgiNbP9YAlR0IcOuiO95Axnn4cBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a2e16a987cf2df4cb03e8926dce9aa12a255da8945084d7318fd77aa58e5e6d5","last_reissued_at":"2026-05-18T00:18:08.774189Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:18:08.774189Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $B$. 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