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A step beyond Kneser’s theorem for Abelian finite groups. (English) Zbl 1032.11009

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.

MSC:

11B13 Additive bases, including sumsets
11B25 Arithmetic progressions
20K01 Finite abelian groups

Citations:

Zbl 0051.28104
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