## 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

Zbl 0051.28104
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