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A $q$-expansion of a number $x$ in $[0,\\frac{1}{q-1}]$ is a sequence $(\\delta_i)_{i=1}^\\infty\\in\\{0,1\\}^{\\mathbb{N}}$ satisfying $$ x=\\sum_{i=1}^\\infty\\frac{\\delta_i}{q^i}.$$ Let $\\mathcal{B}_{\\aleph_0}$ denote the set of $q$ for which there exists $x$ with a countable number of $q$-expansions, and let $\\mathcal{B}_{1, \\aleph_0}$ denote the set of $q$ for which $1$ has a countable number of $q$-expansions. In \\cite{Sidorov6} it was shown that $\\min\\mathcal{B}_{\\aleph_0}=\\min\\mathcal{B}_{1,\\aleph_0}=\\frac{1+\\sqrt{5}}{2},$ and in \\cite{Baker} it was shown that $\\mathcal{B}_{\\ale"},"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":"1502.07212","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-02-25T15:45:24Z","cross_cats_sorted":[],"title_canon_sha256":"3df27474a21676d84a1c4f126cbd98c5aeb38be8b38575686449bd5680b4bcf3","abstract_canon_sha256":"b7b49059cc142a8dc327d642fc6930aa6e995e99a8b56c7292063d126e024626"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:21:55.729392Z","signature_b64":"jV/mfgrLkQF5Fb78gQeFRhe5lUsBO7OYdH8HZNIUxvG6L8KW/6WKJ8RM1cfXNuzp+2jd7UevzG+M0KLksUlACg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"25c17a14d7fdec89829fbcc9ac794e81da1c8c22d7efeaaa564907bc14288a40","last_reissued_at":"2026-05-18T01:21:55.728730Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:21:55.728730Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On small bases for which $1$ has countably many expansions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jian Lu, Lijin Wang, Simon Baker, Yuru Zou","submitted_at":"2015-02-25T15:45:24Z","abstract_excerpt":"Let $q\\in(1,2)$. A $q$-expansion of a number $x$ in $[0,\\frac{1}{q-1}]$ is a sequence $(\\delta_i)_{i=1}^\\infty\\in\\{0,1\\}^{\\mathbb{N}}$ satisfying $$ x=\\sum_{i=1}^\\infty\\frac{\\delta_i}{q^i}.$$ Let $\\mathcal{B}_{\\aleph_0}$ denote the set of $q$ for which there exists $x$ with a countable number of $q$-expansions, and let $\\mathcal{B}_{1, \\aleph_0}$ denote the set of $q$ for which $1$ has a countable number of $q$-expansions. In \\cite{Sidorov6} it was shown that $\\min\\mathcal{B}_{\\aleph_0}=\\min\\mathcal{B}_{1,\\aleph_0}=\\frac{1+\\sqrt{5}}{2},$ and in \\cite{Baker} it was shown that $\\mathcal{B}_{\\ale"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.07212","kind":"arxiv","version":1},"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":"1502.07212","created_at":"2026-05-18T01:21:55.728854+00:00"},{"alias_kind":"arxiv_version","alias_value":"1502.07212v1","created_at":"2026-05-18T01:21:55.728854+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.07212","created_at":"2026-05-18T01:21:55.728854+00:00"},{"alias_kind":"pith_short_12","alias_value":"EXAXUFGX7XWI","created_at":"2026-05-18T12:29:19.899920+00:00"},{"alias_kind":"pith_short_16","alias_value":"EXAXUFGX7XWITAU7","created_at":"2026-05-18T12:29:19.899920+00:00"},{"alias_kind":"pith_short_8","alias_value":"EXAXUFGX","created_at":"2026-05-18T12:29:19.899920+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/EXAXUFGX7XWITAU7XTE2Y6KOQH","json":"https://pith.science/pith/EXAXUFGX7XWITAU7XTE2Y6KOQH.json","graph_json":"https://pith.science/api/pith-number/EXAXUFGX7XWITAU7XTE2Y6KOQH/graph.json","events_json":"https://pith.science/api/pith-number/EXAXUFGX7XWITAU7XTE2Y6KOQH/events.json","paper":"https://pith.science/paper/EXAXUFGX"},"agent_actions":{"view_html":"https://pith.science/pith/EXAXUFGX7XWITAU7XTE2Y6KOQH","download_json":"https://pith.science/pith/EXAXUFGX7XWITAU7XTE2Y6KOQH.json","view_paper":"https://pith.science/paper/EXAXUFGX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1502.07212&json=true","fetch_graph":"https://pith.science/api/pith-number/EXAXUFGX7XWITAU7XTE2Y6KOQH/graph.json","fetch_events":"https://pith.science/api/pith-number/EXAXUFGX7XWITAU7XTE2Y6KOQH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EXAXUFGX7XWITAU7XTE2Y6KOQH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EXAXUFGX7XWITAU7XTE2Y6KOQH/action/storage_attestation","attest_author":"https://pith.science/pith/EXAXUFGX7XWITAU7XTE2Y6KOQH/action/author_attestation","sign_citation":"https://pith.science/pith/EXAXUFGX7XWITAU7XTE2Y6KOQH/action/citation_signature","submit_replication":"https://pith.science/pith/EXAXUFGX7XWITAU7XTE2Y6KOQH/action/replication_record"}},"created_at":"2026-05-18T01:21:55.728854+00:00","updated_at":"2026-05-18T01:21:55.728854+00:00"}