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 \(C\).


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