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)


11B25 Arithmetic progressions
11B83 Special sequences and polynomials
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