{"paper":{"title":"On the projections of Ahlfors regular sets in the plane","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Tuomas Orponen","submitted_at":"2024-10-09T13:34:50Z","abstract_excerpt":"This paper contains the following $\\delta$-discretised projection theorem for Ahlfors regular sets in the plane.\n  For all $C,\\epsilon > 0$ and $s \\in [0,1]$, there exists $\\kappa > 0$ such that the following holds for all $\\delta > 0$ small enough. Let $\\nu$ be a Borel probability measure on $S^{1}$ satisfying $\\nu(B(x,r)) \\leq Cr^{\\epsilon}$ for all $x \\in S^{1}$ and $r > 0$. Let $K \\subset B(1) \\subset \\mathbb{R}^{2}$ be Ahlfors $s$-regular with constant at most $C$. Then, there exists a vector $\\theta \\in \\mathrm{spt\\,} \\nu$ such that $$|\\pi_{\\theta}(F)|_{\\delta} \\geq \\delta^{\\epsilon - s}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2410.06872","kind":"arxiv","version":2},"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/2410.06872/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"}