{"paper":{"title":"$\\alpha$-Expansions with odd partial quotients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Claire Merriman, Florin P. Boca","submitted_at":"2018-06-16T01:38:27Z","abstract_excerpt":"We consider an analogue of Nakada's $\\alpha$-continued fraction transformation in the setting of continued fractions with odd partial quotients. More precisely, given $\\alpha \\in [\\frac{1}{2}(\\sqrt{5}-1),\\frac{1}{2}(\\sqrt{5}+1)]$, we show that every irrational number $x\\in I_\\alpha=[\\alpha-2,\\alpha)$ can be uniquely represented as $$ x= \\cfrac{e_1 (x;\\alpha)}{d_1 (x;\\alpha) +\\cfrac{e_2(x;\\alpha)}{d_2(x;\\alpha)+\\cdots}} , $$ with $e_i(x;\\alpha) \\in \\{ \\pm 1\\}$ and $d_i(x;\\alpha) \\in 2{\\mathbb N} -1$ determined by the iterates of the transformation $$\\varphi_\\alpha (x) := \\frac{1}{| x|} - 2 \\big"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.06166","kind":"arxiv","version":5},"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"}