{"paper":{"title":"Pinned distance problem, slicing measures and local smoothing estimates","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.MG"],"primary_cat":"math.CA","authors_text":"Alex Iosevich, Bochen Liu","submitted_at":"2017-06-29T17:00:18Z","abstract_excerpt":"We improve the Peres-Schlag result on pinned distances in sets of a given Hausdorff dimension. In particular, for Euclidean distances, with $$\\Delta^y(E) = \\{|x-y|:x\\in E\\},$$ we prove that for any $E, F\\subset{\\Bbb R}^d$, there exists a probability measure $\\mu_F$ on $F$ such that for $\\mu_F$-a.e. $y\\in F$,\n  (1) $\\dim_{{\\mathcal H}}(\\Delta^y(E))\\geq\\beta$ if $\\dim_{{\\mathcal H}}(E) + \\frac{d-1}{d+1}\\dim_{{\\mathcal H}}(F) > d - 1 + \\beta$;\n  (2) $\\Delta^y(E)$ has positive Lebesgue measure if $\\dim_{{\\mathcal H}}(E)+\\frac{d-1}{d+1}\\dim_{{\\mathcal H}}(F) > d$;\n  (3) $\\Delta^y(E)$ has non-empty "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.09851","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"}