{"paper":{"title":"On problems of Danzer and Gowers and dynamics on the space of closed subsets of $\\mathbb{R}^d$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.DS","authors_text":"Barak Weiss, Omri Solan, Yaar Solomon","submitted_at":"2015-10-24T19:47:46Z","abstract_excerpt":"Considering the space of closed subsets of $\\mathbb{R}^d$, endowed with the Chabauty-Fell topology, and the affine action of $SL_d(\\mathbb{R})\\ltimes\\mathbb{R}^d$, we prove that the only minimal subsystems are the fixed points $\\{\\varnothing\\}$ and $\\{\\mathbb{R}^d\\}$. As a consequence we resolve a question of Gowers concerning the existence of certain Danzer sets: there is no set $Y \\subset \\mathbb{R}^d$ such that for every convex set $\\mathcal{C} \\subset \\mathbb{R}^d$ of volume one, the cardinality of $\\mathcal{C} \\cap Y$ is bounded above and below by nonzero contants independent of $\\mathcal"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.07179","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"}