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

Green, Ben; Ruzsa, Imre Z.

J. Lond. Math. Soc., II. Ser. 75, No. 1, 163-175 (2007).

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 \(C\).

Reviewer: Mihály Szalay (Budapest)

MSC:

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

Keywords:

inverse problem of additive number theory; Freiman’s theorem; multidimensional arithmetic progressions; Abelian groups; coset progressions

Citations:

Zbl 0203.35305; Zbl 0271.10044; Zbl 0816.11008

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