{"paper":{"title":"Box-counting by H\\\"older's traveling salesman","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.MG","authors_text":"Roger Z\\\"ust, Zolt\\'an Balogh","submitted_at":"2019-07-11T14:25:33Z","abstract_excerpt":"We provide a sufficient Dini-type condition for a subset of a complete, quasiconvex metric space to be covered by a H\\\"older curve. This implies in particular that if the upper box-counting dimension of a set in a quasiconvex metric space is less or equal to $d \\geq 1$, then for any $\\alpha < \\frac{1}{d}$ the set can be covered by an $\\alpha$-H\\\"older curve. On the other hand, for each $1\\leq d <2$ we give an example of a compact set $K$, in the plane, just failing the above Dini-type condition, with lower box-counting dimension equal to zero and upper box-counting dimension equal to $d$ that "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.05227","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"}