{"paper":{"title":"Irrational numbers associated to sequences without geometric progressions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Kevin O'Bryant, Melvyn B. Nathanson","submitted_at":"2013-07-30T20:14:33Z","abstract_excerpt":"Let s and k be integers with s \\geq 2 and k \\geq 2. Let g_k^{(s)}(n) denote the cardinality of the largest subset of the set {1,2,..., n} that contains no geometric progression of length k whose common ratio is a power of s. Let r_k(\\ell) denote the cardinality of the largest subset of the set {0,1,2,\\ldots, \\ell -1\\} that contains no arithmetric progression of length k. The limit \\[ \\lim_{n\\rightarrow \\infty} \\frac{g_k^{(s)}(n)}{n} = (s-1) \\sum_{m=1}^{\\infty}\n  \\left(\\frac{1}{s} \\right)^{\\min \\left(r_k^{-1}(m)\\right)} \\] exists and converges to an irrational number."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.8135","kind":"arxiv","version":1},"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"}