{"paper":{"title":"Intersection properties of typical compact sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Changhao Chen","submitted_at":"2015-12-15T11:11:30Z","abstract_excerpt":"We prove that a typical compact set does not contain any similar copy of a given pattern. We also prove that a typical compact set of $[0,1]^{d} (d\\geq 2)$ intersects any $(d-1)$-dimensional plane in at most $d$ points. We study the \"hitting probabilities\" of compact sets in the sense of Baire category. In the end we study the arithmetic properties of typical compact sets in $[0,1]$ and the \"hitting probabilities\" of continuous functions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.04726","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":""},"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"}