{"paper":{"title":"Falconer distance problem, additive energy and Cartesian products","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.CA","authors_text":"Alex Iosevich, Bochen Liu","submitted_at":"2015-06-25T01:41:08Z","abstract_excerpt":"A celebrated result due to Wolff says if $E$ is a compact subset of ${\\Bbb R}^2$, then the Lebesgue measure of the distance set $\\Delta(E)=\\{|x-y|: x,y \\in E \\}$ is positive if the Hausdorff dimension of $E$ is greater than $\\frac{4}{3}$. In this paper we improve the $\\frac{4}{3}$ barrier by a small exponent for Cartesian products. In higher dimensions, also in the context of Cartesian products, we reduce Erdogan's $\\frac{d}{2}+\\frac{1}{3}$ exponent to $\\frac{d^2}{2d-1}$. The proof uses a combination of Fourier analysis and additive comibinatorics."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.07595","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"}