{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:WQ33LJRENXLC4JLRTPC4MLIBMI","short_pith_number":"pith:WQ33LJRE","schema_version":"1.0","canonical_sha256":"b437b5a6246dd62e25719bc5c62d01622653a9b59081a6c6f69f747823208a4a","source":{"kind":"arxiv","id":"1504.04789","version":2},"attestation_state":"computed","paper":{"title":"Restrictions of H\\\"older continuous functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.PR","authors_text":"Andr\\'as M\\'ath\\'e, Omer Angel, Rich\\'ard Balka, Yuval Peres","submitted_at":"2015-04-19T04:26:09Z","abstract_excerpt":"For $0<\\alpha<1$ let $V(\\alpha)$ denote the supremum of the numbers $v$ such that every $\\alpha$-H\\\"older continuous function is of bounded variation on a set of Hausdorff dimension $v$. Kahane and Katznelson (2009) proved the estimate $1/2 \\leq V(\\alpha)\\leq 1/(2-\\alpha)$ and asked whether the upper bound is sharp. We show that in fact $V(\\alpha)=\\max\\{1/2,\\alpha\\}$. Let $\\dim_{H}$ and $\\overline{\\dim}_{M}$ denote the Hausdorff and upper Minkowski dimension, respectively. The upper bound on $V(\\alpha)$ is a consequence of the following theorem. Let $\\{B(t): t\\in [0,1]\\}$ be a fractional Brown"},"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":"1504.04789","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-04-19T04:26:09Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"a65c09442249b5b19d1e88398b540def324f2e5a9a562e0f3a1b3f326d23b18b","abstract_canon_sha256":"abc145b198d341d569cf6b9e151be121b2f534cd6fb1155ef210fc78a952ae52"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:56:38.259992Z","signature_b64":"z5JUx9zlIkkxsbQpAt8SWS/ModYk+kXPwxjHaTvX89hyNOmhn2ED4+BqvIC35lRaEUpbl/TCPEQLOjqlQx8LCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b437b5a6246dd62e25719bc5c62d01622653a9b59081a6c6f69f747823208a4a","last_reissued_at":"2026-05-18T00:56:38.259252Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:56:38.259252Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Restrictions of H\\\"older continuous functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.PR","authors_text":"Andr\\'as M\\'ath\\'e, Omer Angel, Rich\\'ard Balka, Yuval Peres","submitted_at":"2015-04-19T04:26:09Z","abstract_excerpt":"For $0<\\alpha<1$ let $V(\\alpha)$ denote the supremum of the numbers $v$ such that every $\\alpha$-H\\\"older continuous function is of bounded variation on a set of Hausdorff dimension $v$. Kahane and Katznelson (2009) proved the estimate $1/2 \\leq V(\\alpha)\\leq 1/(2-\\alpha)$ and asked whether the upper bound is sharp. We show that in fact $V(\\alpha)=\\max\\{1/2,\\alpha\\}$. Let $\\dim_{H}$ and $\\overline{\\dim}_{M}$ denote the Hausdorff and upper Minkowski dimension, respectively. The upper bound on $V(\\alpha)$ is a consequence of the following theorem. Let $\\{B(t): t\\in [0,1]\\}$ be a fractional Brown"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.04789","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":"1504.04789","created_at":"2026-05-18T00:56:38.259367+00:00"},{"alias_kind":"arxiv_version","alias_value":"1504.04789v2","created_at":"2026-05-18T00:56:38.259367+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.04789","created_at":"2026-05-18T00:56:38.259367+00:00"},{"alias_kind":"pith_short_12","alias_value":"WQ33LJRENXLC","created_at":"2026-05-18T12:29:47.479230+00:00"},{"alias_kind":"pith_short_16","alias_value":"WQ33LJRENXLC4JLR","created_at":"2026-05-18T12:29:47.479230+00:00"},{"alias_kind":"pith_short_8","alias_value":"WQ33LJRE","created_at":"2026-05-18T12:29:47.479230+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/WQ33LJRENXLC4JLRTPC4MLIBMI","json":"https://pith.science/pith/WQ33LJRENXLC4JLRTPC4MLIBMI.json","graph_json":"https://pith.science/api/pith-number/WQ33LJRENXLC4JLRTPC4MLIBMI/graph.json","events_json":"https://pith.science/api/pith-number/WQ33LJRENXLC4JLRTPC4MLIBMI/events.json","paper":"https://pith.science/paper/WQ33LJRE"},"agent_actions":{"view_html":"https://pith.science/pith/WQ33LJRENXLC4JLRTPC4MLIBMI","download_json":"https://pith.science/pith/WQ33LJRENXLC4JLRTPC4MLIBMI.json","view_paper":"https://pith.science/paper/WQ33LJRE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1504.04789&json=true","fetch_graph":"https://pith.science/api/pith-number/WQ33LJRENXLC4JLRTPC4MLIBMI/graph.json","fetch_events":"https://pith.science/api/pith-number/WQ33LJRENXLC4JLRTPC4MLIBMI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WQ33LJRENXLC4JLRTPC4MLIBMI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WQ33LJRENXLC4JLRTPC4MLIBMI/action/storage_attestation","attest_author":"https://pith.science/pith/WQ33LJRENXLC4JLRTPC4MLIBMI/action/author_attestation","sign_citation":"https://pith.science/pith/WQ33LJRENXLC4JLRTPC4MLIBMI/action/citation_signature","submit_replication":"https://pith.science/pith/WQ33LJRENXLC4JLRTPC4MLIBMI/action/replication_record"}},"created_at":"2026-05-18T00:56:38.259367+00:00","updated_at":"2026-05-18T00:56:38.259367+00:00"}