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Let $\\mathcal{U}=\\mathcal{U}(M)$ be the set of \\emph{univoque bases} $q>1$ for which $1$ has a unique $q$-expansion.\n  The main object of this paper is to provide new characterizations of $\\mathcal{U}$ and to show that the Hausdorff dimension of the set of numbers $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":"1606.03791","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2016-06-13T02:00:54Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"3d7996d72ae864b7d06125d2d31588291b6f1e65a2817eb954005871413c56aa","abstract_canon_sha256":"b9be3efd78531ac8cdff57442c0d77d8bb56642cbd0c959c97d99bc9704d968a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:47:27.103704Z","signature_b64":"iinTxaUxIOlkjYyrDSYMqkwvWzPWlp8AkhgIUb1rGEYznXQy295xMEWAWYKslHOtfvdHHhhM2xt2587yaUpiAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5defc05d9001e17d66a0580c8e97e94c65843d365336071775ad20ac41eb37c8","last_reissued_at":"2026-05-18T00:47:27.103038Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:47:27.103038Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Univoque bases and Hausdorff dimension","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.NT","authors_text":"Derong Kong, Fan L\\\"u, Martijn de Vries, Wenxia Li","submitted_at":"2016-06-13T02:00:54Z","abstract_excerpt":"Given a positive integer $M$ and a real number $q >1$, a \\emph{$q$-expansion} of a real number $x$ is a sequence $(c_i)=c_1c_2\\cdots$ with $(c_i) \\in \\{0,\\ldots,M\\}^\\infty$ such that \\[x=\\sum_{i=1}^{\\infty} c_iq^{-i}.\\]\n  It is well known that if $q \\in (1,M+1]$, then each $x \\in I_q:=\\left[0,M/(q-1)\\right]$ has a $q$-expansion. Let $\\mathcal{U}=\\mathcal{U}(M)$ be the set of \\emph{univoque bases} $q>1$ for which $1$ has a unique $q$-expansion.\n  The main object of this paper is to provide new characterizations of $\\mathcal{U}$ and to show that the Hausdorff dimension of the set of numbers $x \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.03791","kind":"arxiv","version":2},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1606.03791","created_at":"2026-05-18T00:47:27.103130+00:00"},{"alias_kind":"arxiv_version","alias_value":"1606.03791v2","created_at":"2026-05-18T00:47:27.103130+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.03791","created_at":"2026-05-18T00:47:27.103130+00:00"},{"alias_kind":"pith_short_12","alias_value":"LXX4AXMQAHQX","created_at":"2026-05-18T12:30:29.479603+00:00"},{"alias_kind":"pith_short_16","alias_value":"LXX4AXMQAHQX2ZVA","created_at":"2026-05-18T12:30:29.479603+00:00"},{"alias_kind":"pith_short_8","alias_value":"LXX4AXMQ","created_at":"2026-05-18T12:30:29.479603+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/LXX4AXMQAHQX2ZVALAGI5F7JJR","json":"https://pith.science/pith/LXX4AXMQAHQX2ZVALAGI5F7JJR.json","graph_json":"https://pith.science/api/pith-number/LXX4AXMQAHQX2ZVALAGI5F7JJR/graph.json","events_json":"https://pith.science/api/pith-number/LXX4AXMQAHQX2ZVALAGI5F7JJR/events.json","paper":"https://pith.science/paper/LXX4AXMQ"},"agent_actions":{"view_html":"https://pith.science/pith/LXX4AXMQAHQX2ZVALAGI5F7JJR","download_json":"https://pith.science/pith/LXX4AXMQAHQX2ZVALAGI5F7JJR.json","view_paper":"https://pith.science/paper/LXX4AXMQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1606.03791&json=true","fetch_graph":"https://pith.science/api/pith-number/LXX4AXMQAHQX2ZVALAGI5F7JJR/graph.json","fetch_events":"https://pith.science/api/pith-number/LXX4AXMQAHQX2ZVALAGI5F7JJR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LXX4AXMQAHQX2ZVALAGI5F7JJR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LXX4AXMQAHQX2ZVALAGI5F7JJR/action/storage_attestation","attest_author":"https://pith.science/pith/LXX4AXMQAHQX2ZVALAGI5F7JJR/action/author_attestation","sign_citation":"https://pith.science/pith/LXX4AXMQAHQX2ZVALAGI5F7JJR/action/citation_signature","submit_replication":"https://pith.science/pith/LXX4AXMQAHQX2ZVALAGI5F7JJR/action/replication_record"}},"created_at":"2026-05-18T00:47:27.103130+00:00","updated_at":"2026-05-18T00:47:27.103130+00:00"}