{"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"}