zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Generalized arithmetical progressions and sumsets. (English) Zbl 0816.11008
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 $\prod\sb{i=1}\sp d (\ell\sb i+ 1)$. 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 $Cn$. 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)

11B25Arithmetic progressions
11B83Special sequences of integers and polynomials
Full Text: DOI
[1] N. N. Bogolyubov, Some algebraical properties of almost periods (in Russian),Zap. kafedry mat. fiziki Kiev,4 (1939), 185--194. · Zbl 65.0268.01
[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). · Zbl 0271.10044
[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. · Zbl 0728.11009
[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