{"paper":{"title":"Furstenberg sets for a fractal set of directions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Ezequiel Rela, Ursula Molter","submitted_at":"2010-09-02T17:23:23Z","abstract_excerpt":"In this note we study the behavior of the size of Furstenberg sets with respect to the size of the set of directions defining it. For any pair $\\alpha,\\beta\\in(0,1]$, we will say that a set $E\\subset \\R^2$ is an $F_{\\alpha\\beta}$-set if there is a subset $L$ of the unit circle of Hausdorff dimension at least $\\beta$ and, for each direction $e$ in $L$, there is a line segment $\\ell_e$ in the direction of $e$ such that the Hausdorff dimension of the set $E\\cap\\ell_e$ is equal or greater than $\\alpha$. The problem is considered in the wider scenario of generalized Hausdorff measures, giving estim"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.0481","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"}