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A subset $S$ of $\\mathcal{F}$ is $k$-product-free if no element of $S$ can be obtained by concatenating $k$ words from $S$, and strongly $k$-product-free if no element of $S$ is a (non-trivial) concatenation of at most $k$ words from $S$.\n  We prove that a $k$-product-free subset of $\\mathcal{F}$ has upper Banach density at most $1/\\rho(k)$, where $\\rho(k) = \\min\\{\\ell \\colon \\ell \\nmid k - 1\\}$. We also determine the"},"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":"2305.05304","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2023-05-09T09:48:28Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"594d945b31e6d81bc4f553880ebb5ce600210033569b3e3d037fe67432436237","abstract_canon_sha256":"a1bda6fd60a15a25c55f79c23561d5449a82839ac5a25289ed875ac5bf21c55b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:28:36.726742Z","signature_b64":"AvY4hCTJ64qXXJbFxHNcRdTZUPqMVrfEOhDaaMfXbBZYwEdydPNT0+FwOO7q7ladZmfyxBfWnhhVhtmmoHvIAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"140847b38765322009ba7cddc41eb19efe91e4eafad8c5381072250a3eb52f57","last_reissued_at":"2026-07-05T06:28:36.726157Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:28:36.726157Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The structure and density of $k$-product-free sets in the free semigroup","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.CO","authors_text":"Alex Scott, Freddie Illingworth, Lukas Michel","submitted_at":"2023-05-09T09:48:28Z","abstract_excerpt":"The free semigroup $\\mathcal{F}$ over a finite alphabet $\\mathcal{A}$ is the set of all finite words with letters from $\\mathcal{A}$ equipped with the operation of concatenation. A subset $S$ of $\\mathcal{F}$ is $k$-product-free if no element of $S$ can be obtained by concatenating $k$ words from $S$, and strongly $k$-product-free if no element of $S$ is a (non-trivial) concatenation of at most $k$ words from $S$.\n  We prove that a $k$-product-free subset of $\\mathcal{F}$ has upper Banach density at most $1/\\rho(k)$, where $\\rho(k) = \\min\\{\\ell \\colon \\ell \\nmid k - 1\\}$. We also determine the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2305.05304","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2305.05304/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2305.05304","created_at":"2026-07-05T06:28:36.726224+00:00"},{"alias_kind":"arxiv_version","alias_value":"2305.05304v2","created_at":"2026-07-05T06:28:36.726224+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2305.05304","created_at":"2026-07-05T06:28:36.726224+00:00"},{"alias_kind":"pith_short_12","alias_value":"CQEEPM4HMUZC","created_at":"2026-07-05T06:28:36.726224+00:00"},{"alias_kind":"pith_short_16","alias_value":"CQEEPM4HMUZCACN2","created_at":"2026-07-05T06:28:36.726224+00:00"},{"alias_kind":"pith_short_8","alias_value":"CQEEPM4H","created_at":"2026-07-05T06:28:36.726224+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/CQEEPM4HMUZCACN2PTO4IHVRT3","json":"https://pith.science/pith/CQEEPM4HMUZCACN2PTO4IHVRT3.json","graph_json":"https://pith.science/api/pith-number/CQEEPM4HMUZCACN2PTO4IHVRT3/graph.json","events_json":"https://pith.science/api/pith-number/CQEEPM4HMUZCACN2PTO4IHVRT3/events.json","paper":"https://pith.science/paper/CQEEPM4H"},"agent_actions":{"view_html":"https://pith.science/pith/CQEEPM4HMUZCACN2PTO4IHVRT3","download_json":"https://pith.science/pith/CQEEPM4HMUZCACN2PTO4IHVRT3.json","view_paper":"https://pith.science/paper/CQEEPM4H","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2305.05304&json=true","fetch_graph":"https://pith.science/api/pith-number/CQEEPM4HMUZCACN2PTO4IHVRT3/graph.json","fetch_events":"https://pith.science/api/pith-number/CQEEPM4HMUZCACN2PTO4IHVRT3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CQEEPM4HMUZCACN2PTO4IHVRT3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CQEEPM4HMUZCACN2PTO4IHVRT3/action/storage_attestation","attest_author":"https://pith.science/pith/CQEEPM4HMUZCACN2PTO4IHVRT3/action/author_attestation","sign_citation":"https://pith.science/pith/CQEEPM4HMUZCACN2PTO4IHVRT3/action/citation_signature","submit_replication":"https://pith.science/pith/CQEEPM4HMUZCACN2PTO4IHVRT3/action/replication_record"}},"created_at":"2026-07-05T06:28:36.726224+00:00","updated_at":"2026-07-05T06:28:36.726224+00:00"}