## Freiman’s theorem in an arbitrary Abelian group.(English)Zbl 1133.11058

According to a famous result of G. A. Freiman [Foundations of a structural theory of set addition, Kazan (1966; Zbl 0203.35305) and Providence, R. I.: AMS (1973; Zbl 0271.10044)], if $$A$$ is a finite set of integers and $$|A+ A|\leq K|A|$$, then $$A$$ is contained in a proper arithmetic progression of dimension at most $$d(K)$$ and size at most $$f(K)|A|$$. (Recall that $$P$$ is a proper arithmetic progression of dimension $$d$$ and size $$L$$ if $$P= \{v_0+\ell_1 v_1+\cdots+ \ell_d v_d: 0\leq \ell_j< L_j\}$$ and $$L_1L_2\cdots L_d= L= |P|$$.)
A different proof of Freiman’s theorem was given by the second author [Acta Math. Hung. 65, No. 4, 379–388 (1994; Zbl 0816.11008)]. In the present paper, the authors follow the broad scheme of this proof and prove the following analogous theorem for Abelian groups. Let $$G$$ be an arbitrary Abelian group and let $$A$$ be a finite subset of $$G$$ with $$|A+ A|\leq K|A|$$. Then $$A$$ is contained in a coset progression $$P+H$$ where $$H$$ is a subgroup of $$G$$, $$P$$ is a proper arithmetic progression of dimension at most $$CK^4\log(K+ 2)$$ and size at most $$\exp(CK^4\log^2(K+ 2))|A|$$ with an absolute constant $$C$$.

### MSC:

 11P70 Inverse problems of additive number theory, including sumsets 11B99 Sequences and sets 11B75 Other combinatorial number theory

### Citations:

Zbl 0203.35305; Zbl 0271.10044; Zbl 0816.11008
Full Text:

### References:

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