{"paper":{"title":"Hausdorff dimension of pinned distance sets and the $L^2$-method","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.MG"],"primary_cat":"math.CA","authors_text":"Bochen Liu","submitted_at":"2018-10-18T16:01:45Z","abstract_excerpt":"We prove that for any $E\\subset{\\Bbb R}^2$, $\\dim_{\\mathcal{H}}(E)>1$, there exists $x\\in E$ such that the Hausdorff dimension of the pinned distance set\n  $$\\Delta_x(E)=\\{|x-y|: y \\in E\\}$$\n  is no less than $\\min\\left\\{\\frac{4}{3}\\dim_{\\mathcal{H}}(E)-\\frac{2}{3}, 1\\right\\}$. This answers a question recently raised by Guth, Iosevich, Ou and Wang, as well as improves results of Keleti and Shmerkin.\n  (This version is already published on Proceeding AMS so I would like to leave it unchanged. However the statement in the abstract, which is the second part of Theorem 1.1, should be weakened a bi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.08127","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1810.08127/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}