{"paper":{"title":"Restrictions of Brownian motion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.PR","authors_text":"Rich\\'ard Balka, Yuval Peres","submitted_at":"2014-06-11T06:12:47Z","abstract_excerpt":"Let $\\{ B(t) \\colon 0\\leq t\\leq 1\\}$ be a linear Brownian motion and let $\\dim$ denote the Hausdorff dimension. Let $\\alpha>\\frac12$ and $1\\leq \\beta \\leq 2$. We prove that, almost surely, there exists no set $A\\subset[0,1]$ such that $\\dim A>\\frac12$ and $B\\colon A\\to\\mathbb{R}$ is $\\alpha$-H\\\"older continuous. The proof is an application of Kaufman's dimension doubling theorem. As a corollary of the above theorem, we show that, almost surely, there exists no set $A\\subset[0,1]$ such that $\\dim A>\\frac{\\beta}{2}$ and $B\\colon A\\to\\mathbb{R}$ has finite $\\beta$-variation. The zero set of $B$ a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.2789","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"}