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Generalized arithmetical progressions and sumsets. (English) Zbl 0816.11008

Let $a,{q}_{1},\cdots ,{q}_{d}$ be elements of an arbitrary commutative group and let ${\ell }_{1},\cdots ,{\ell }_{d}$ be positive integers. A set of the form

$P\left({q}_{1},\cdots ,{q}_{d};{\ell }_{1},\cdots ,{\ell }_{d};a\right)=\left\{n=a+{x}_{1}{q}_{1}+\cdots +{x}_{d}{q}_{d},\phantom{\rule{4pt}{0ex}}0\le {x}_{i}\le {\ell }_{i}\right\}$

is called a $d$-dimensional generalized arithmetic progression. Its size is defined to be the quantity ${\prod }_{i=1}^{d}\left({\ell }_{i}+1\right)$.

The author proves the following theorem: Let $A$, $B$ be finite sets in a torsionfree commutative group with $|A|=|B|=n$ and $|A+B|\le \alpha n$. 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 of integers and polynomials
##### References:
 [1] N. N. Bogolyubov, Some algebraical properties of almost periods (in Russian),Zap. kafedry mat. fiziki Kiev,4 (1939), 185–194. [2] J. Bourgain, On arithmetic progressions in sums of sets of integers, in:A tribute to Paul Erdos, eds. A. Baker, B. Bollobás, A. Hajnal, Cambridge Univ. Press (Cambridge, England, 1990), pp. 105–109. [3] G. A. Freiman,Foundations of a Structural Theory of Set Addition (in Russian), Kazan Gos. Ped. Inst. (Kazan, 1966). [4] G. A. Freiman,Foundations of a Structural Theory of Set Addition, Translation of Mathematical Monographs vol. 37, Amer. Math. Soc. (Providence, R. I., USA, 1973). [5] G. A. Freiman, What is the structure ofK ifK+K is small?, in:Lecture Notes in Mathematics 1240 Springer-Verlag, (New York-Berlin, 1987), pp. 109–134. [6] G. A. Freiman, H. Halberstam and I. Z. Ruzsa, Integer sum sets containing long arithmetic progressions,J. London Math. Soc.,46 (1992), 193–201. · Zbl 0768.11005 · doi:10.1112/jlms/s2-46.2.193 [7] I. Z. Ruzsa, Arithmetic progressions in sumsets,Acta Arithmetica,60 (1991), 191–202. [8] I. Z. Ruzsa, Arithmetical progressions and the number of sums,Periodica Math. Hung.,25 (1992), 105–111. · Zbl 0761.11005 · doi:10.1007/BF02454387