{"paper":{"title":"An inverse theorem for an inequality of Kneser","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Terence Tao","submitted_at":"2017-11-12T18:43:15Z","abstract_excerpt":"Let $G = (G,+)$ be a compact connected abelian group, and let $\\mu_G$ denote its probability Haar measure. A theorem of Kneser (generalising previous results of Macbeath and Raikov) establishes the bound $$ \\mu_G(A + B) \\geq \\min( \\mu_G(A)+\\mu_G(B), 1 ) $$ whenever $A,B$ are compact subsets of $G$, and $A+B := \\{ a+b: a \\in A, b \\in B \\}$ denotes the sumset of $A$ and $B$. Clearly one has equality when $\\mu_G(A)+\\mu_G(B) \\geq 1$. Another way in which equality can be obtained is when $A = \\phi^{-1}(I), B = \\phi^{-1}(J)$ for some continuous surjective homomorphism $\\phi: G \\to {\\bf R}/{\\bf Z}$ a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.04337","kind":"arxiv","version":4},"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"}