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For $k=2, 3,\\cdots,\\aleph_0$ let $\\mathcal{B}_k$ be the set of bases $q$ for which there exists $x$ having $k$ different $q$-expansions, and for $q\\in \\mathcal{B}_k$ let $\\mathcal{U}_q^{(k)}$ be the set of all such $x$'s which have $k$ different $q$-expansions. In this paper we show that \\[ \\mathcal{B}_{\\aleph_0}=[2,\\infty),\\quad \\mathcal{B}_k=(q_c,\\infty)\\quad \\textrm{for any}\\quad k\\ge 2, \\] where $q_c\\approx 2.32472$ is the appropriate root of"},"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":"1508.06138","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-08-25T13:13:58Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"e3737c040382d99d02233568b7d445a2e8c337c46f02d7fbfda97b182b655a2b","abstract_canon_sha256":"912f45f8a6aec9332f4a38d3a311c2dc6399681865a17cc22be5854a4a9f8262"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:10:19.758559Z","signature_b64":"9qi8WVNDtpNHmKOaWW4cPM7QIROjCLds/Oyyy3/fAzAzYhVT3UDIQe0OzIEa+u7XAZH/6PwCQ7TZz5px3MI3Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fad080e432e4e2d0a56e8f8c163a49275c1b3038d41d821771d0abce56790b1e","last_reissued_at":"2026-05-18T00:10:19.757968Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:10:19.757968Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Multiple expansions of real numbers with digits set $\\{0,1,q\\}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.NT","authors_text":"Derong Kong, Kan Jiang, Karma Dajani, Wenxia Li","submitted_at":"2015-08-25T13:13:58Z","abstract_excerpt":"For $q>1$ we consider expansions in base $q$ over the alphabet $\\{0,1,q\\}$. 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