{"paper":{"title":"Pinned Dot Product Set Estimates","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CA","authors_text":"Caleb Marshall, Paige Bright, Steven Senger","submitted_at":"2024-12-23T21:10:44Z","abstract_excerpt":"We study a variant of the Falconer distance problem for dot products. In particular, for fractal subsets $A\\subset \\mathbb{R}^n$ and $a,x\\in \\mathbb{R}^n$, we study sets of the form \\[ \\Pi_x^a(A) := \\{\\alpha \\in \\mathbb{R} : (a-x)\\cdot y= \\alpha, \\text{ for some $y\\in A$}\\}. \\] We discuss some of what is already known to give a picture of the current state of the art, as well as prove some new results and special cases. We obtain lower bounds on the Hausdorff dimension of $A$ to guarantee that $\\Pi^a_x(A)$ is large in some quantitative sense for some $a\\in A$ (i.e. $\\Pi_x^a(A)$ has large Hausd"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.17985","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2412.17985/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"}