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Zbl 0816.11008
Ruzsa, I.Z.
Generalized arithmetical progressions and sumsets.
(English)
[J] Acta Math. Hung. 65, No.4, 379-388 (1994). ISSN 0236-5294; ISSN 1588-2632/e

Let $a, q\sb 1,\dots, q\sb d$ be elements of an arbitrary commutative group and let $\ell\sb 1, \dots, \ell\sb d$ be positive integers. A set of the form $$P(q\sb 1,\dots, q\sb d; \ell\sb 1,\dots, \ell\sb d; a)=\{n= a+x\sb 1 q\sb 1+\cdots+ x\sb d q\sb d,\ 0\leq x\sb i\leq \ell\sb i\}$$ is called a $d$-dimensional generalized arithmetic progression. Its size is defined to be the quantity $\prod\sb{i=1}\sp d (\ell\sb i+ 1)$. \par The author proves the following theorem: Let $A$, $B$ be finite sets in a torsionfree commutative group with $\vert A\vert=\vert B\vert=n$ and $\vert A+ B\vert\leq \alpha n$. Then there are numbers $d$ and $C$ depending only on $\alpha$ such that $A$ is contained in a generalized arithmetic progression of dimension at most $d$ and of size at most $Cn$. \par This result, in the author's opinion, is essentially equivalent to a famous theorem of Freiman although it is expressed in different terms and the proof is along completely different lines.
[M.Nair (Glasgow)]
MSC 2000:
*11B25 Arithmetic progressions
11B83 Special sequences of integers and polynomials

Keywords: sumsets; Freiman theorem; generalized arithmetic progression; torsionfree commutative group

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