{"paper":{"title":"Invariant games and non-homogeneous Beatty sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Eric Duch\\^ene, Julien Cassaigne, Michel Rigo","submitted_at":"2013-12-08T16:23:44Z","abstract_excerpt":"We characterize all the pairs of complementary non-homogenous Beatty sequences $(A_n)_{n\\ge 0}$ and $(B_n)_{n\\ge 0}$ for which there exists an invariant game having exactly $\\{(A_n,B_n)\\mid n\\ge 0\\}\\cup \\{(B_n,A_n)\\mid n\\ge 0\\}$ as set of $\\mathcal{P}$-positions. Using the notion of Sturmian word and tools arising in symbolic dynamics and combinatorics on words, this characterization can be translated to a decision procedure relying only on a few algebraic tests about algebraicity or rational independence. Given any four real numbers defining the two sequences, up to these tests, we can theref"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.2233","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"}