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Combinatorics and linear algebra of Freiman's isomorphism

Part of: Sequences and sets

Published online by Cambridge University Press:  26 February 2010

Sergei V. Konyagin  and
Vsevolod F. Lev
Sergei V. Konyagin
Affiliation:
Department of Mechanics and Mathematics, Moscow State University, Moscow, 119899, Russia. E-mail: kon@nw.math.msu.su
Vsevolod F. Lev
Affiliation:
Department of Mathematics, The University of Georgia, Athens, GA 30602, U.S.A.
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Abstract

An original linear algebraic approach to the basic notion of Freiman's isomorphism is developed and used in conjunction with a combinatorial argument to answer two questions, posed by Freiman about 35 years ago.

First, the order of growth is established of t(n), the number of classes isomorphic n-element sets of integers: t(n) = n(2 + σ(1))n. Second, it is proved linear Roth sets (sets of integers free of arithmetic progressions and having Freiman rank 1) exist and, moreover, the number of classes of such cardinality n is amazingly large; in fact, it is “the same as above”: .

MSC classification

Secondary: 11B13: Additive bases, including sumsets
Type
Research Article
Information
Mathematika , Volume 47 , Issue 1-2 , December 2000 , pp. 39 - 51
Copyright
Copyright © University College London 2000

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References

1.Freiman, G.. Foundations of a structure theory of set addition, Kazan 1966 [Russian]. English translation in: Translations of Math. Monographs 37 (1973), American Math. Soc., Providence, RI.Google Scholar
2.Asaphova, G. A.. On one type of Latin squares of order 6, in: Number-theoretic investigations in Markov spectrum and structure theory of set addition, Kalinin State University, Moscow, 1973 [Russian].Google Scholar