{"paper":{"title":"Dimension of Pinned Distance Sets for Semi-Regular Sets","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.LO"],"primary_cat":"math.CA","authors_text":"D. M. Stull, Jacob B. Fiedler","submitted_at":"2023-09-21T00:58:35Z","abstract_excerpt":"We prove that if $E\\subseteq \\R^2$ is analytic and $1<d < \\dim_H(E)$, there are ``many'' points $x\\in E$ such that the Hausdorff dimension of the pinned distance set $\\Delta_x E$ is at least $d\\left(1 - \\frac{\\left(D-1\\right)\\left(D-d\\right)}{2D^2+\\left(2-4d\\right)D+d^2+d-2}\\right)$, where $D = \\dim_P(E)$. In particular, we prove that $\\dim_H(\\Delta_x E) \\geq \\frac{d(d-4)}{d-5}$ for these $x$, which gives the best known lower bound for this problem when $d \\in (1, 5-\\sqrt{15})$. We also prove that there exists some $x\\in E$ such that the packing dimension of $\\Delta_x E$ is at least $\\frac{12 "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2309.11701","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/2309.11701/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"}