{"paper":{"title":"Power Laws for the Favard Length Problem in $\\mathbb{R}^d$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.MG","math.NT"],"primary_cat":"math.CA","authors_text":"Caleb Marshall","submitted_at":"2025-09-02T23:05:43Z","abstract_excerpt":"We prove a power law for the asymptotic decay of the Favard length of neighbourhoods of certain self-similar sets in $\\mathbb{R}^d$ with $d \\geq 2$. These self-similar sets are generalizations of the so-called four-corner Cantor set to higher dimensions, as well as to a more general class of rational digit sets. When $d \\geq 3$, our estimates are the first such non-trivial asymptotic upper bounds for the Favard length problem. The extension to a new class of digit sets (which is new even when $d = 2$, but holds for $d \\geq 2$ generally) uses the work of G. Kiss, I. Laba, G. Somlai and the auth"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.02882","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/2509.02882/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"}