{"paper":{"title":"Matching for generalised $\\beta$-transformations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Carlo Carminati, Charlene Kalle, Henk Bruin","submitted_at":"2016-10-06T13:49:53Z","abstract_excerpt":"We investigate matching for the family $T_\\alpha(x) = \\beta x + \\alpha \\pmod 1$, $\\alpha \\in [0,1]$, for fixed $\\beta > 1$. Matching refers to the property that there is an $n \\in \\mathbb N$ such that $T_\\alpha^n(0) = T_\\alpha^n(1)$. We show that for various Pisot numbers $\\beta$, matching occurs on an open dense set of $\\alpha \\in [0,1]$ and we compute the Hausdorff dimension of its complement. Numerical evidence shows more cases where matching is prevalent."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.01872","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"}