{"paper":{"title":"Reconstructing geometric objects from the measures of their intersections with test sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Andr\\'as M\\'ath\\'e, M\\'arton Elekes, Tam\\'as Keleti","submitted_at":"2011-09-28T11:18:09Z","abstract_excerpt":"Let us say that an element of a given family $\\A$ of subsets of $\\R^d$ can be reconstructed using $n$ test sets if there exist $T_1,...,T_n \\subset \\R^d$ such that whenever $A,B\\in \\A$ and the Lebesgue measures of $A \\cap T_i$ and $B \\cap T_i$ agree for each $i=1,...,n$ then $A=B$. Our goal will be to find the least such $n$.\n  We prove that if $\\A$ consists of the translates of a fixed reasonably nice subset of $\\R^d$ then this minimum is $n=d$. In order to obtain this result we reconstruct a translate of a fixed function using $d$ test sets as well, and also prove that under rather mild cond"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.6169","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"}