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# A note on Freiman models in Heisenberg groups

(Impact Factor: 0.79). 01/2010; 189(1):1-15. DOI: 10.1007/s11856-011-0175-5

ABSTRACT

Green and Ruzsa recently proved that for any s ≥ 2, any small squaring set A in a (multiplicative) abelian group, i.e., |A·A| < K|A|, has a Freiman smodel: it means that there exists a group G and a Freiman s-isomorphism from A into G such that |G| < f (s,K)|A|.
In an unpublished note, Green proved that such a result does not necessarily hold in nonabelian groups if s ≥ 64. The aim of this paper is improve Green’s result by showing that it remains true under the weaker assumption s ≥ 6.

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Available from: Francois Hennecart, Nov 01, 2014
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• "The lower bound in the theorem is also proved to be sharp up to a constant factor, in the sense that there is a set of points P and a set of lines L with |P| = |L| = q 3/2 that determines no incidence (for details see [34]). Theorem 1.1 has various applications in several combinatorial number theory problems (for example [13] [14] [19]). "
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