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Specifically we show the following.\n  For every integer $k\\geq 2$ and every set $A$ of words over $k$ satisfying \\[\\limsup_{n\\to\\infty} \\frac{|A\\cap [k]^n|}{k^n}>0\\] there exist a word $c$ over $k$ and a sequence $(w_n)$ of left variable words over $k$ such that the set \\[\\{c\\}\\cup \\big\\{c^{\\smallfrown}w_0(a_0)^{\\smallfrown}...^{\\smallfrown}w_n(a_n) : n\\in\\mathbb{N} \\ \\text{ and } \\ a_0,...,a_n\\in [k]\\big\\}\\] is contained in $A$.\n  While the result is infinite-dimensional its proof is based on an appropriate finite and quantitative ve"},"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":"1209.4985","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-09-22T11:34:41Z","cross_cats_sorted":[],"title_canon_sha256":"8d4e4639127d1b9097d30fc0b33d3f4009409cc56ca855bf7376584e2e5112f3","abstract_canon_sha256":"79c7936ad5f125dd6960f45963a84656399e53bbd3a4cf9d06a92bccdd6ea5a5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:32:42.118652Z","signature_b64":"41mC18lO00hA0SAQ1lcHL+b0+vkgLvpgswwFtggvxEIpoYsauoat/vJDJixpE9UL2GVA9ifvwfdpxAMbsP54CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"66458bf01f8d8e217759266def948344160a782abb41b2a9d5956ce648d82439","last_reissued_at":"2026-05-18T01:32:42.118223Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:32:42.118223Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A density version of the Carlson--Simpson theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Konstantinos Tyros, Pandelis Dodos, Vassilis Kanellopoulos","submitted_at":"2012-09-22T11:34:41Z","abstract_excerpt":"We prove a density version of the Carlson--Simpson Theorem. 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