## Generalized arithmetical progressions and sumsets.(English)Zbl 0816.11008

Let $$a, q_ 1,\dots, q_ d$$ be elements of an arbitrary commutative group and let $$\ell_ 1, \dots, \ell_ d$$ be positive integers. A set of the form $P(q_ 1,\dots, q_ d; \ell_ 1,\dots, \ell_ d; a)=\{n= a+x_ 1 q_ 1+\cdots+ x_ d q_ d,\;0\leq x_ i\leq \ell_ i\}$ is called a $$d$$-dimensional generalized arithmetic progression. Its size is defined to be the quantity $$\prod_{i=1}^ d (\ell_ i+ 1)$$.
The author proves the following theorem: Let $$A$$, $$B$$ be finite sets in a torsionfree commutative group with $$| A|=| B|=n$$ and $$| A+ B|\leq \alpha n$$. 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 $$Cn$$.
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.
Reviewer: M.Nair (Glasgow)

### MSC:

 11B25 Arithmetic progressions 11B83 Special sequences and polynomials
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### References:

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