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For integers $1 \\leq k <\\ell$ and a compact abelian group $G$, let $$\\lambda_{k,\\ell}(G)=\\sup\\{ \\mu(A): kA \\cap \\ell A =\\emptyset \\}$$ be the maximum possible size of a $(k,\\ell)$-sum-free subset of $G$. We prove that if $G=\\mathbb{I} \\times M$, where $\\mathbb{I}$ is the identity component of $G$, then $$\\lambda_{k, \\ell}(G)=\\max \\left\\{ \\lambda_{k, \\el"},"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":"1901.03233","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-01-10T15:52:01Z","cross_cats_sorted":["math.GR","math.NT"],"title_canon_sha256":"98d640d8a16b3a90fd2e04aee59df868a1df9c08e275d0975eadf84e8a97e525","abstract_canon_sha256":"010de0d045a8cfccfce771ebf72ada2b63c81afd54ca93e9ddcd54a9dc93b832"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:56:22.686464Z","signature_b64":"Zqo5JBuWd6nhpjs+HscRJNKBRJGShEXfJ8c1JwPfnLxOUrrGXTT+FLG2VjQeFU8PVT0t7WrnBGz7cn6aAmVtCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8cb1394da578fa3be8bb741aa54421b5b81ed79064511c5b95a126faac5d6b59","last_reissued_at":"2026-05-17T23:56:22.685766Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:56:22.685766Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The largest $(k, \\ell)$-sum-free sets in compact abelian groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR","math.NT"],"primary_cat":"math.CO","authors_text":"Noah Kravitz","submitted_at":"2019-01-10T15:52:01Z","abstract_excerpt":"A subset $A$ of a finite abelian group is called $(k,\\ell)$-sum-free if $kA \\cap \\ell A=\\emptyset.$ In this paper, we extend this concept to compact abelian groups and study the question of how large a measurable $(k,\\ell)$-sum-free set can be. For integers $1 \\leq k <\\ell$ and a compact abelian group $G$, let $$\\lambda_{k,\\ell}(G)=\\sup\\{ \\mu(A): kA \\cap \\ell A =\\emptyset \\}$$ be the maximum possible size of a $(k,\\ell)$-sum-free subset of $G$. We prove that if $G=\\mathbb{I} \\times M$, where $\\mathbb{I}$ is the identity component of $G$, then $$\\lambda_{k, \\ell}(G)=\\max \\left\\{ \\lambda_{k, \\el"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.03233","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":"1901.03233","created_at":"2026-05-17T23:56:22.685891+00:00"},{"alias_kind":"arxiv_version","alias_value":"1901.03233v2","created_at":"2026-05-17T23:56:22.685891+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.03233","created_at":"2026-05-17T23:56:22.685891+00:00"},{"alias_kind":"pith_short_12","alias_value":"RSYTSTNFPD5D","created_at":"2026-05-18T12:33:27.125529+00:00"},{"alias_kind":"pith_short_16","alias_value":"RSYTSTNFPD5DX2F3","created_at":"2026-05-18T12:33:27.125529+00:00"},{"alias_kind":"pith_short_8","alias_value":"RSYTSTNF","created_at":"2026-05-18T12:33:27.125529+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/RSYTSTNFPD5DX2F3OQNKKRBBWW","json":"https://pith.science/pith/RSYTSTNFPD5DX2F3OQNKKRBBWW.json","graph_json":"https://pith.science/api/pith-number/RSYTSTNFPD5DX2F3OQNKKRBBWW/graph.json","events_json":"https://pith.science/api/pith-number/RSYTSTNFPD5DX2F3OQNKKRBBWW/events.json","paper":"https://pith.science/paper/RSYTSTNF"},"agent_actions":{"view_html":"https://pith.science/pith/RSYTSTNFPD5DX2F3OQNKKRBBWW","download_json":"https://pith.science/pith/RSYTSTNFPD5DX2F3OQNKKRBBWW.json","view_paper":"https://pith.science/paper/RSYTSTNF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1901.03233&json=true","fetch_graph":"https://pith.science/api/pith-number/RSYTSTNFPD5DX2F3OQNKKRBBWW/graph.json","fetch_events":"https://pith.science/api/pith-number/RSYTSTNFPD5DX2F3OQNKKRBBWW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RSYTSTNFPD5DX2F3OQNKKRBBWW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RSYTSTNFPD5DX2F3OQNKKRBBWW/action/storage_attestation","attest_author":"https://pith.science/pith/RSYTSTNFPD5DX2F3OQNKKRBBWW/action/author_attestation","sign_citation":"https://pith.science/pith/RSYTSTNFPD5DX2F3OQNKKRBBWW/action/citation_signature","submit_replication":"https://pith.science/pith/RSYTSTNFPD5DX2F3OQNKKRBBWW/action/replication_record"}},"created_at":"2026-05-17T23:56:22.685891+00:00","updated_at":"2026-05-17T23:56:22.685891+00:00"}