{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:VI3JKHNBUWXQVLAUTIDX5PGOJ2","short_pith_number":"pith:VI3JKHNB","schema_version":"1.0","canonical_sha256":"aa36951da1a5af0aac149a077ebcce4e8417200596650b37a6111df357efc555","source":{"kind":"arxiv","id":"1803.07338","version":1},"attestation_state":"computed","paper":{"title":"The $\\beta$-transformation with a hole at 0","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Charlene Kalle, Derong Kong, Niels Langeveld, Wenxia Li","submitted_at":"2018-03-20T09:52:19Z","abstract_excerpt":"For $\\beta\\in(1,2]$ the $\\beta$-transformation $T_\\beta: [0,1) \\to [0,1)$ is defined by $T_\\beta ( x) = \\beta x \\pmod 1$. For $t\\in[0, 1)$ let $K_\\beta(t)$ be the survivor set of $T_\\beta$ with hole $(0,t)$ given by \\[K_\\beta(t):=\\{x\\in[0, 1): T_\\beta^n(x)\\not \\in (0, t) \\textrm{ for all }n\\ge 0\\}.\\] In this paper we characterise the bifurcation set $E_\\beta$ of all parameters $t\\in[0,1)$ for which the set valued function $t\\mapsto K_\\beta(t)$ is not locally constant. We show that $E_\\beta$ is a Lebesgue null set of full Hausdorff dimension for all $\\beta\\in(1,2)$. We prove that for Lebesgue a"},"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":"1803.07338","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2018-03-20T09:52:19Z","cross_cats_sorted":[],"title_canon_sha256":"e3200d5dc8f433e74d379ed75ff3720f0a3d2b0438714eeec7bd10397d957f50","abstract_canon_sha256":"05df6ad5d4add46365c71bd0ec09ff1bb44fce0dae8be71b6f6a7755a247a180"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:20:33.601688Z","signature_b64":"0/bkiVNgSuYe/kdlGm0xQ+KN0xnhanQPhVvn4DBT+rU9WRlqocC5Zk5JqyqYuT3P1maWQj33LXWBELRxaIlAAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aa36951da1a5af0aac149a077ebcce4e8417200596650b37a6111df357efc555","last_reissued_at":"2026-05-18T00:20:33.601229Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:20:33.601229Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The $\\beta$-transformation with a hole at 0","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Charlene Kalle, Derong Kong, Niels Langeveld, Wenxia Li","submitted_at":"2018-03-20T09:52:19Z","abstract_excerpt":"For $\\beta\\in(1,2]$ the $\\beta$-transformation $T_\\beta: [0,1) \\to [0,1)$ is defined by $T_\\beta ( x) = \\beta x \\pmod 1$. For $t\\in[0, 1)$ let $K_\\beta(t)$ be the survivor set of $T_\\beta$ with hole $(0,t)$ given by \\[K_\\beta(t):=\\{x\\in[0, 1): T_\\beta^n(x)\\not \\in (0, t) \\textrm{ for all }n\\ge 0\\}.\\] In this paper we characterise the bifurcation set $E_\\beta$ of all parameters $t\\in[0,1)$ for which the set valued function $t\\mapsto K_\\beta(t)$ is not locally constant. We show that $E_\\beta$ is a Lebesgue null set of full Hausdorff dimension for all $\\beta\\in(1,2)$. We prove that for Lebesgue a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.07338","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":"1803.07338","created_at":"2026-05-18T00:20:33.601289+00:00"},{"alias_kind":"arxiv_version","alias_value":"1803.07338v1","created_at":"2026-05-18T00:20:33.601289+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.07338","created_at":"2026-05-18T00:20:33.601289+00:00"},{"alias_kind":"pith_short_12","alias_value":"VI3JKHNBUWXQ","created_at":"2026-05-18T12:32:59.047623+00:00"},{"alias_kind":"pith_short_16","alias_value":"VI3JKHNBUWXQVLAU","created_at":"2026-05-18T12:32:59.047623+00:00"},{"alias_kind":"pith_short_8","alias_value":"VI3JKHNB","created_at":"2026-05-18T12:32:59.047623+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/VI3JKHNBUWXQVLAUTIDX5PGOJ2","json":"https://pith.science/pith/VI3JKHNBUWXQVLAUTIDX5PGOJ2.json","graph_json":"https://pith.science/api/pith-number/VI3JKHNBUWXQVLAUTIDX5PGOJ2/graph.json","events_json":"https://pith.science/api/pith-number/VI3JKHNBUWXQVLAUTIDX5PGOJ2/events.json","paper":"https://pith.science/paper/VI3JKHNB"},"agent_actions":{"view_html":"https://pith.science/pith/VI3JKHNBUWXQVLAUTIDX5PGOJ2","download_json":"https://pith.science/pith/VI3JKHNBUWXQVLAUTIDX5PGOJ2.json","view_paper":"https://pith.science/paper/VI3JKHNB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1803.07338&json=true","fetch_graph":"https://pith.science/api/pith-number/VI3JKHNBUWXQVLAUTIDX5PGOJ2/graph.json","fetch_events":"https://pith.science/api/pith-number/VI3JKHNBUWXQVLAUTIDX5PGOJ2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VI3JKHNBUWXQVLAUTIDX5PGOJ2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VI3JKHNBUWXQVLAUTIDX5PGOJ2/action/storage_attestation","attest_author":"https://pith.science/pith/VI3JKHNBUWXQVLAUTIDX5PGOJ2/action/author_attestation","sign_citation":"https://pith.science/pith/VI3JKHNBUWXQVLAUTIDX5PGOJ2/action/citation_signature","submit_replication":"https://pith.science/pith/VI3JKHNBUWXQVLAUTIDX5PGOJ2/action/replication_record"}},"created_at":"2026-05-18T00:20:33.601289+00:00","updated_at":"2026-05-18T00:20:33.601289+00:00"}