Deshouillers, Jean-Marc; Freiman, Gregory A. A step beyond Kneser’s theorem for Abelian finite groups. (English) Zbl 1032.11009 Proc. Lond. Math. Soc. (3) 86, No. 1, 1-28 (2003). The main result of this note is the following Theorem 1: Let \(n\in \mathbb{N}\) and \({\mathcal A}\) be a nonempty subset of \(\mathbb{Z}/n\mathbb{Z}\) with \(|{\mathcal A}|\leq cn\) (for some absolute constant \(c\)) which satisfies \(|{\mathcal A}+{\mathcal A}|\leq 2.04|{\mathcal A}|\). Then there exists a proper subgroup \({\mathcal H}\) of \(\mathbb{Z}/n\mathbb{Z}\) such that one of the following three cases holds: (i) if the number of cosets met by \({\mathcal A}\), let us call it \(s\), is different from 1 and 3, then \({\mathcal A}\) is included in an arithmetic progression of \(\ell\) cosets modulo \({\mathcal H}\) such that \[ (\ell-1)|{\mathcal H}|\leq|{\mathcal A}+{\mathcal A}|-|{\mathcal A}|; \tag{1} \] (ii) if \({\mathcal A}\) meets exactly three cosets modulo \({\mathcal H}\), that is, if \(s=3\), then (1) holds with \(\ell\) replaced by \(\min(\ell,4)\);(iii) if \({\mathcal A}\) is included in a single coset modulo \({\mathcal H}\), then we have \(|{\mathcal A}|> c|{\mathcal H}|\). This is shown as a generalization of M. Kneser’s theorem [Math. Z. 58, 459–484 (1953; Zbl 0051.28104)]. The proof of Theorem 1 distinguishes several cases. Reviewer: E.Härtter (Mainz) Cited in 1 ReviewCited in 9 Documents MSC: 11B13 Additive bases, including sumsets 11B25 Arithmetic progressions 20K01 Finite abelian groups Citations:Zbl 0051.28104 PDF BibTeX XML Cite \textit{J.-M. Deshouillers} and \textit{G. A. Freiman}, Proc. Lond. Math. Soc. (3) 86, No. 1, 1--28 (2003; Zbl 1032.11009) Full Text: DOI OpenURL