2011/06/11 by Terence Tao, Tao, Terence
Mathematics · #11B30 #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Limits and Structures in Graph Theory #math.CO #msc:11B30
paper · pdf · doi:10.48550/arxiv.1106.2267
8 pages, no figures. To appear, European Journal of Combinatorics. This is the final version, incorporating the referee corrections
openalex publication_date 2011/06/11 · arxiv created 2012/04/03 · arxiv updated 2012/04/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A corollary of Kneser's theorem, one sees that any finite non-empty subset A of an abelian group G = (G,+) with |A + A| ≤ (2-\eps) |A| can be covered by at most (2)/(\eps)-1 translates of a finite group H of cardinality at most (2-\eps)|A|. Using some arguments of Hamidoune, we establish an analogue in the noncommutative setting. Namely, if A is a finite non-empty subset of a nonabelian group G = (G,⋅) such that |A ⋅ A| ≤ (2-\eps) |A|, then A is either contained in a right-coset of a finite group H of cardinality at most (2)/(\eps)|A|, or can be covered by at most (2)/(\eps)-1 right-cosets of a finite group H of cardinality at most |A|. We also note some connections with some recent work of Sanders and of Petridis.