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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
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References:

[1] Bilu, Structure of sets with small sumset, in: Structure theory of set addition pp 77– (1999) · Zbl 0946.11004
[2] Bogolyubov, Sur quelques propriétés arithmétiques des presque-périodes, Ann. Chaire Math. Phys. Kiev 4 pp 185– (1939)
[3] Cassels, An introduction to the geometry of numbers, in: Classics in mathematics (1997) · Zbl 0866.11041
[4] Chang, A polynomial bound in Freiman’s theorem, Duke Math. J. 113 pp 399– (2002) · Zbl 1035.11048
[5] Deshouillers, On small sumsets in (\(\mathbb{Z}\)/2\(\mathbb{Z}\))n, Combinatorica 24 pp 53– (2004) · Zbl 1049.11108
[6] Freiman, Foundations of a structural theory of set addition 37 (1973) · Zbl 0271.10044
[7] B. J. Green Edinburgh-MIT lecture notes on Freiman’s theorem 2006 http://www.dpmms.cam.ac.uk/ bjg23
[8] Ruzsa, Sets with small sumset and rectification, Bull. London Math. Soc. 38 pp 43– (2006) · Zbl 1155.11307
[9] Plünnecke, Eigenschaften un Abschätzungen von Wirkingsfunktionen (1969)
[10] Rudin, Interscience Tracts in Pure and Applied Mathematics 12, in: Fourier analysis on groups (1962)
[11] Ruzsa, On the cardinality of A + A and A - A pp 933– (1978)
[12] Ruzsa, An application of graph theory to additive number theory, Sci. Ser. A Math. Sci. (N.S.) 3 pp 97– (1989) · Zbl 0743.05052
[13] Ruzsa, Generalized arithmetical progressions and sumsets, Acta Math. Hungar. 65 pp 379– (1994) · Zbl 0816.11008
[14] Ruzsa, An analog of Freiman’s theorem in groups, in: Structure theory of set addition pp 323– (1999)
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