{"paper":{"title":"Group actions, the Mattila integral and applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.CA","authors_text":"Bochen Liu","submitted_at":"2017-05-01T15:21:52Z","abstract_excerpt":"The Mattila integral, $$ {\\mathcal M}(\\mu)=\\int {\\left( \\int_{S^{d-1}} {|\\widehat{\\mu}(r \\omega)|}^2 d\\omega \\right)}^2 r^{d-1} dr,$$ developed by Mattila, is the main tool in the study of the Falconer distance problem. In this paper, with a very simple argument, we develop a generalized version of the Mattila integral. Our first application is to consider the product of distances $$(\\Delta(E))^k= \\left\\{\\prod_{j=1}^k |x^j-y^j|: x^j, y^j\\in E\\right\\} $$ and show that when $d\\geq 2$, $(\\Delta(E))^k$ has positive Lebesgue measure if $\\dim_{\\mathcal{H}}(E)>\\frac{d}{2}+\\frac{1}{4k-1}$. Another app"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.00560","kind":"arxiv","version":4},"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"}