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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1806.06166","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2018-06-16T01:38:27Z","cross_cats_sorted":[],"title_canon_sha256":"14aeef8617c7df59f516f9422d0c8b17e60e54bd279a0798562e0b1bc9c383e6","abstract_canon_sha256":"6923018475948d5f4bbcdc2d677b30ae308f4acf3400717042413485c46fb254"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:41:46.475354Z","signature_b64":"4/EVhdTqsZeeL9nBJoqyGc+0hyVBsJksJS4V3xogbd9ZzGcIqke1qC75jZ9nbV7l1ZCNM+kn/IqLenjz4kvhCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a7ba7322be4fc7ac0764e8e22c1a0adfee323583d68f65a836cfda798fc45cfc","last_reissued_at":"2026-05-17T23:41:46.474669Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:41:46.474669Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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