{"paper":{"title":"Functional van den Berg-Kesten-Reimer Inequalities and their Duals, with Applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Larry Goldstein, Yosef Rinott","submitted_at":"2015-08-28T16:30:08Z","abstract_excerpt":"The BKR inequality conjectured by van den Berg and Kesten in [11], and proved by Reimer in [8], states that for $A$ and $B$ events on $S$, a finite product of finite sets $S_i,i=1,\\ldots,n$, and $P$ any product measure on $S$, $$ P(A \\Box B) \\le P(A)P(B),$$ where the set $A \\Box B$ consists of the elementary events which lie in both $A$ and $B$ for `disjoint reasons.' Precisely, with ${\\bf n}:=\\{1,\\ldots,n\\}$ and $K \\subset {\\bf n}$, for ${\\bf x} \\in S$ letting $[{\\bf x}]_K=\\{{\\bf y} \\in S: y_i = x_i, i \\in K\\}$, the set $A \\Box B$ consists of all ${\\bf x} \\in S$ for which there exist disjoint"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.07267","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"}