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Let $\\mathbf{U}_{{\\beta},N}$ be the set of all $x$'s in $\\Gamma_{{\\beta},N}$ which have unique ${\\beta}$-expansions. We give explicit formula of the Hausdorff dimension of $\\mathbf{U}_{{\\beta},N}$ for ${\\beta}$ in any admissible interval $[{{\\beta}}_L,{{\\beta}}_U]$, where ${{\\beta}_L}$ is a purely Parry number while ${{\\beta}_U}$ is a tra"},"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":"1401.6473","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.DS","submitted_at":"2014-01-24T23:50:29Z","cross_cats_sorted":[],"title_canon_sha256":"59b7dd3a1bacab7a6fedd65d4085a6fded6be23f5a7e7de78f2abae5fe60c459","abstract_canon_sha256":"b1ceb8b1351a8efb28eb24f610b3723e93058b4635b33ada9383927106c517d7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:02.116125Z","signature_b64":"FhVVi+OP13CvykrxDQKJd8LHjbOtkVee9gzcdGo9cAvajbAwTFCLh0vAu0h4KyT6qPfWF6e1ktqn3idhl10IBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8d130440809d5e35740d4ce280565e1cc2e8a36123c502f51ff5a565de61b5fb","last_reissued_at":"2026-05-18T01:36:02.115271Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:02.115271Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Hausdorff dimension of unique beta expansions","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Derong Kong, Wenxia Li","submitted_at":"2014-01-24T23:50:29Z","abstract_excerpt":"Given an integer $N\\ge 2$ and a real number ${\\beta}>1$, let $\\Gamma_{{\\beta},N}$ be the set of all $x=\\sum_{i=1}^\\infty {d_i}/{{\\beta}^i}$ with $d_i\\in\\{0,1,\\cdots,N-1\\}$ for all $i\\ge 1$. The infinite sequence $(d_i)$ is called a ${\\beta}$-expansion of $x$. Let $\\mathbf{U}_{{\\beta},N}$ be the set of all $x$'s in $\\Gamma_{{\\beta},N}$ which have unique ${\\beta}$-expansions. We give explicit formula of the Hausdorff dimension of $\\mathbf{U}_{{\\beta},N}$ for ${\\beta}$ in any admissible interval $[{{\\beta}}_L,{{\\beta}}_U]$, where ${{\\beta}_L}$ is a purely Parry number while ${{\\beta}_U}$ is a tra"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.6473","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":"1401.6473","created_at":"2026-05-18T01:36:02.115437+00:00"},{"alias_kind":"arxiv_version","alias_value":"1401.6473v2","created_at":"2026-05-18T01:36:02.115437+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1401.6473","created_at":"2026-05-18T01:36:02.115437+00:00"},{"alias_kind":"pith_short_12","alias_value":"RUJQIQEATVPD","created_at":"2026-05-18T12:28:46.137349+00:00"},{"alias_kind":"pith_short_16","alias_value":"RUJQIQEATVPDK5AN","created_at":"2026-05-18T12:28:46.137349+00:00"},{"alias_kind":"pith_short_8","alias_value":"RUJQIQEA","created_at":"2026-05-18T12:28:46.137349+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/RUJQIQEATVPDK5ANJTRIAVS6DT","json":"https://pith.science/pith/RUJQIQEATVPDK5ANJTRIAVS6DT.json","graph_json":"https://pith.science/api/pith-number/RUJQIQEATVPDK5ANJTRIAVS6DT/graph.json","events_json":"https://pith.science/api/pith-number/RUJQIQEATVPDK5ANJTRIAVS6DT/events.json","paper":"https://pith.science/paper/RUJQIQEA"},"agent_actions":{"view_html":"https://pith.science/pith/RUJQIQEATVPDK5ANJTRIAVS6DT","download_json":"https://pith.science/pith/RUJQIQEATVPDK5ANJTRIAVS6DT.json","view_paper":"https://pith.science/paper/RUJQIQEA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1401.6473&json=true","fetch_graph":"https://pith.science/api/pith-number/RUJQIQEATVPDK5ANJTRIAVS6DT/graph.json","fetch_events":"https://pith.science/api/pith-number/RUJQIQEATVPDK5ANJTRIAVS6DT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RUJQIQEATVPDK5ANJTRIAVS6DT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RUJQIQEATVPDK5ANJTRIAVS6DT/action/storage_attestation","attest_author":"https://pith.science/pith/RUJQIQEATVPDK5ANJTRIAVS6DT/action/author_attestation","sign_citation":"https://pith.science/pith/RUJQIQEATVPDK5ANJTRIAVS6DT/action/citation_signature","submit_replication":"https://pith.science/pith/RUJQIQEATVPDK5ANJTRIAVS6DT/action/replication_record"}},"created_at":"2026-05-18T01:36:02.115437+00:00","updated_at":"2026-05-18T01:36:02.115437+00:00"}