{"paper":{"title":"Small Furstenberg sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.CA","authors_text":"Ezequiel Rela, Ursula Molter","submitted_at":"2010-06-24T19:22:36Z","abstract_excerpt":"For $\\alpha$ in $(0,1]$, a subset $E$ of $\\RR$ is called Furstenberg set of type $\\alpha$ or $F_\\alpha$-set if for each direction $e$ in the unit circle 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 greater or equal than $\\alpha$. In this paper we show that if $\\alpha > 0$, there exists a set $E\\in F_\\alpha$ such that $\\HH{g}(E)=0$ for $g(x)=x^{1/2+3/2\\alpha}\\log^{-\\theta}(\\frac{1}{x})$, $\\theta>\\frac{1+3\\alpha}{2}$, which improves on the the previously known bound, that $H^{\\beta}(E) = 0$ for $\\beta>1/2+3/2\\alpha$. Furth"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1006.4862","kind":"arxiv","version":3},"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"}