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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","authors_text":"Konstantinos Tyros, Pandelis Dodos, Vassilis Kanellopoulos","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-09-22T11:34:41Z","title":"A density version of the Carlson--Simpson theorem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.4985","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:41f114e786ad68b6e5382077d92a1e89e06542846ac236fc5ca46687f6f281df","target":"record","created_at":"2026-05-18T01:32:42Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"79c7936ad5f125dd6960f45963a84656399e53bbd3a4cf9d06a92bccdd6ea5a5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-09-22T11:34:41Z","title_canon_sha256":"8d4e4639127d1b9097d30fc0b33d3f4009409cc56ca855bf7376584e2e5112f3"},"schema_version":"1.0","source":{"id":"1209.4985","kind":"arxiv","version":3}},"canonical_sha256":"66458bf01f8d8e217759266def948344160a782abb41b2a9d5956ce648d82439","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"66458bf01f8d8e217759266def948344160a782abb41b2a9d5956ce648d82439","first_computed_at":"2026-05-18T01:32:42.118223Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:32:42.118223Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"41mC18lO00hA0SAQ1lcHL+b0+vkgLvpgswwFtggvxEIpoYsauoat/vJDJixpE9UL2GVA9ifvwfdpxAMbsP54CA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:32:42.118652Z","signed_message":"canonical_sha256_bytes"},"source_id":"1209.4985","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:41f114e786ad68b6e5382077d92a1e89e06542846ac236fc5ca46687f6f281df","sha256:d35eb420577906db74c0c3691417d9cd9c6d90448cdb8b2ef5293ebe56c98444"],"state_sha256":"5e233bea622fa75a11bb7b49d38770b2590c670fc41ce2449c2ef86b67aa34a8"}