{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:HCW27KDRMVYK3PG5O6WRWWJVN4","short_pith_number":"pith:HCW27KDR","schema_version":"1.0","canonical_sha256":"38adafa8716570adbcdd77ad1b59356f1808c210ed805ec4747ee24ee4e6507d","source":{"kind":"arxiv","id":"1604.00747","version":1},"attestation_state":"computed","paper":{"title":"A dichotomy law for the Diophantine properties in $\\beta$-dynamical systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.NT","authors_text":"Bao-Wei Wang, Michael Coons, Mumtaz Hussain","submitted_at":"2016-04-04T06:05:45Z","abstract_excerpt":"Let $\\beta>1$ be a real number and define the $\\beta$-transformation on $[0,1]$ by $T_\\beta:x\\mapsto \\beta x\\bmod 1$. Further, define $$W_y(T_{\\beta},\\Psi):=\\{x\\in [0, 1]:|T_\\beta^nx-y|<\\Psi(n) \\mbox{ for infinitely many $n$}\\}$$ and $$W(T_{\\beta},\\Psi):=\\{(x, y)\\in [0, 1]^2:|T_\\beta^nx-y|<\\Psi(n) \\mbox{ for infinitely many $n$}\\},$$ where $\\Psi:\\mathbb{N}\\to\\mathbb{R}_{>0}$ is a positive function such that $\\Psi(n)\\to 0$ as $n\\to \\infty$. In this paper, we show that each of the above sets obeys a Jarn\\'ik-type dichotomy, that is, the generalised Hausdorff measure is either zero or full depend"},"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":"1604.00747","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-04-04T06:05:45Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"e384b0017cbe5d2adf8cf7109e6a56321e0ef84b071a1b4247913d0922be88fe","abstract_canon_sha256":"299cc8051ecc36a27b144fc00843324c5f88e74b48c88921187488a03c2c9f2a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:13:53.278307Z","signature_b64":"BygDqF+X48J/DAsT0YNcnTmw3qNA/wCW0btyBojVUp8ufBGS3I57/oogxt/NdyUO5hmr+oLvhtCIhAVlVfroDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"38adafa8716570adbcdd77ad1b59356f1808c210ed805ec4747ee24ee4e6507d","last_reissued_at":"2026-05-18T01:13:53.277779Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:13:53.277779Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A dichotomy law for the Diophantine properties in $\\beta$-dynamical systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.NT","authors_text":"Bao-Wei Wang, Michael Coons, Mumtaz Hussain","submitted_at":"2016-04-04T06:05:45Z","abstract_excerpt":"Let $\\beta>1$ be a real number and define the $\\beta$-transformation on $[0,1]$ by $T_\\beta:x\\mapsto \\beta x\\bmod 1$. Further, define $$W_y(T_{\\beta},\\Psi):=\\{x\\in [0, 1]:|T_\\beta^nx-y|<\\Psi(n) \\mbox{ for infinitely many $n$}\\}$$ and $$W(T_{\\beta},\\Psi):=\\{(x, y)\\in [0, 1]^2:|T_\\beta^nx-y|<\\Psi(n) \\mbox{ for infinitely many $n$}\\},$$ where $\\Psi:\\mathbb{N}\\to\\mathbb{R}_{>0}$ is a positive function such that $\\Psi(n)\\to 0$ as $n\\to \\infty$. In this paper, we show that each of the above sets obeys a Jarn\\'ik-type dichotomy, that is, the generalised Hausdorff measure is either zero or full depend"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.00747","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":"1604.00747","created_at":"2026-05-18T01:13:53.277854+00:00"},{"alias_kind":"arxiv_version","alias_value":"1604.00747v1","created_at":"2026-05-18T01:13:53.277854+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.00747","created_at":"2026-05-18T01:13:53.277854+00:00"},{"alias_kind":"pith_short_12","alias_value":"HCW27KDRMVYK","created_at":"2026-05-18T12:30:19.053100+00:00"},{"alias_kind":"pith_short_16","alias_value":"HCW27KDRMVYK3PG5","created_at":"2026-05-18T12:30:19.053100+00:00"},{"alias_kind":"pith_short_8","alias_value":"HCW27KDR","created_at":"2026-05-18T12:30:19.053100+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/HCW27KDRMVYK3PG5O6WRWWJVN4","json":"https://pith.science/pith/HCW27KDRMVYK3PG5O6WRWWJVN4.json","graph_json":"https://pith.science/api/pith-number/HCW27KDRMVYK3PG5O6WRWWJVN4/graph.json","events_json":"https://pith.science/api/pith-number/HCW27KDRMVYK3PG5O6WRWWJVN4/events.json","paper":"https://pith.science/paper/HCW27KDR"},"agent_actions":{"view_html":"https://pith.science/pith/HCW27KDRMVYK3PG5O6WRWWJVN4","download_json":"https://pith.science/pith/HCW27KDRMVYK3PG5O6WRWWJVN4.json","view_paper":"https://pith.science/paper/HCW27KDR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1604.00747&json=true","fetch_graph":"https://pith.science/api/pith-number/HCW27KDRMVYK3PG5O6WRWWJVN4/graph.json","fetch_events":"https://pith.science/api/pith-number/HCW27KDRMVYK3PG5O6WRWWJVN4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HCW27KDRMVYK3PG5O6WRWWJVN4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HCW27KDRMVYK3PG5O6WRWWJVN4/action/storage_attestation","attest_author":"https://pith.science/pith/HCW27KDRMVYK3PG5O6WRWWJVN4/action/author_attestation","sign_citation":"https://pith.science/pith/HCW27KDRMVYK3PG5O6WRWWJVN4/action/citation_signature","submit_replication":"https://pith.science/pith/HCW27KDRMVYK3PG5O6WRWWJVN4/action/replication_record"}},"created_at":"2026-05-18T01:13:53.277854+00:00","updated_at":"2026-05-18T01:13:53.277854+00:00"}