{"paper":{"title":"Measurable Steinhaus sets do not exist for finite sets or the integers in the plane","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.MG","authors_text":"Michael Papadimitrakis, Mihail N. Kolountzakis","submitted_at":"2016-04-21T18:40:41Z","abstract_excerpt":"A Steinhaus set $S \\subseteq \\RR^d$ for a set $A \\subseteq \\RR^d$ is a set such that $S$ has exactly one point in common with $\\tau A$, for every rigid motion $\\tau$ of $\\RR^d$. We show here that if $A$ is a finite set of at least two points then there is no such set $S$ which is Lebesgue measurable.\n  An old result of Komj\\'ath says that there exists a Steinhaus set for $A = \\ZZ\\times\\Set{0}$ in $\\RR^2$. We also show here that such a set cannot be Lebesgue measurable."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.06454","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":""},"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"}