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It is known that almost every $x$ belonging to the interval $[0,m/(q-1)]$ has uncountably many expansions. In this paper we study the existence of expansions $(d_i)$ of $x$ satisfying the inequalities $\\sum_{i=1}^n d_iq^{-i} \\geq \\sum_{i=1}^n c_i q^{-i}$, $n=1,2,...$ for each expansion $(c_i)$ of $x$."},"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":"1011.5220","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-11-23T20:59:35Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"602ea31564c3a241df0bc0e3de3e205835754924417782346ce8b063e6a9bddb","abstract_canon_sha256":"4ab793dd439f6a891738392cf31892876ad9df287142f1afda37071168952f40"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:22:02.226380Z","signature_b64":"uH8HI1olWtXVwfzXPb196b9U43I6KLbIgVTfmw+wSJl0UtRAP0Jut125j1EWjjh/aXlxPUJXo0xtbiwfUEX2Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d0c5559578cc0e4c496d7b081b9fb149963e692eabe599c8dd0bada5d02201f5","last_reissued_at":"2026-05-18T04:22:02.225951Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:22:02.225951Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Optimal expansions in non-integer bases","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.NT","authors_text":"Karma Dajani, Martijn de Vries, Paola Loreti, Vilmos Komornik","submitted_at":"2010-11-23T20:59:35Z","abstract_excerpt":"For a given positive integer $m$, let $A=\\set{0,1,...,m}$ and $q \\in (m,m+1)$. 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