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Combinatorics and linear algebra of Freiman's isomorphism
Author(s) -
Konyagin Sergei V.,
Lev Vsevolod F.
Publication year - 2000
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300015709
Subject(s) - mathematics , cardinality (data modeling) , isomorphism (crystallography) , rank (graph theory) , order (exchange) , combinatorics , algebraic number , element (criminal law) , linear algebra , discrete mathematics , algebra over a field , pure mathematics , mathematical analysis , chemistry , finance , computer science , political science , crystal structure , law , economics , data mining , crystallography , geometry
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 numbert 1 ′ ( n )of classes of such cardinality n is amazingly large; in fact, it is “the same as above”:t 1 ′ ( n ) = n ( 2 + o ( 1 ) ) n.