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Given any $x\\in[0, (\\beta-1)^{-1}]$, a sequence $(a_n)\\in\\{0,1\\}^{\\mathbb{N}}$ is called a $\\beta$-expansion of $x$ if $x=\\sum_{n=1}^{\\infty}a_n\\beta^{-n}.$ For any $k\\geq 1$ and any $(b_1b_2\\cdots b_k)\\in\\{0,1\\}^{k}$, if there exists some $k_0$ such that $a_{k_0+1}a_{k_0+2}\\cdots a_{k_0+k}=b_1b_2\\cdots b_k$, then we call $(a_n)$ a universal $\\beta$-expansion of $x$.\n  Sidorov \\cite{Sidorov2003}, Dajani and de Vries \\cite{DajaniDeVrie} proved that given any $1<\\beta<2$, then Lebesgue almost every point has uncountably many universal expansions. In this paper we consider the se"},"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":"1703.02172","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2017-03-07T01:39:49Z","cross_cats_sorted":["math.MG","math.NT"],"title_canon_sha256":"8acceeb3ce1b83c99b429c25ce9e0da5c31bee7cea0f021bdaae52e642289773","abstract_canon_sha256":"9b235b903a460db8b88e7845b7ad61c5c4b6efe49b97ca5f1f90ef5c21ae146b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:49:20.559692Z","signature_b64":"tcqtyS1AabovGTFe04PqI3HzGjtYxw3/7U5XFcyWPZ3cZJitZNAcp0KSmMOyVWIq1rCIQuBbygbAcGg21QMMCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a0a4c7bae6b4738bda7ef9f11ced1ce35aa23a037c81d78f6bbe52de90eb785f","last_reissued_at":"2026-05-18T00:49:20.559331Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:49:20.559331Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the points without universal expansions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG","math.NT"],"primary_cat":"math.DS","authors_text":"Kan Jiang, Karma Dajani","submitted_at":"2017-03-07T01:39:49Z","abstract_excerpt":"Let $1<\\beta<2$. Given any $x\\in[0, (\\beta-1)^{-1}]$, a sequence $(a_n)\\in\\{0,1\\}^{\\mathbb{N}}$ is called a $\\beta$-expansion of $x$ if $x=\\sum_{n=1}^{\\infty}a_n\\beta^{-n}.$ For any $k\\geq 1$ and any $(b_1b_2\\cdots b_k)\\in\\{0,1\\}^{k}$, if there exists some $k_0$ such that $a_{k_0+1}a_{k_0+2}\\cdots a_{k_0+k}=b_1b_2\\cdots b_k$, then we call $(a_n)$ a universal $\\beta$-expansion of $x$.\n  Sidorov \\cite{Sidorov2003}, Dajani and de Vries \\cite{DajaniDeVrie} proved that given any $1<\\beta<2$, then Lebesgue almost every point has uncountably many universal expansions. 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