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Ensembles rectifiables (d'après Seva Lev)

Le : 22/03/2007 14h15
Par : Yu Bilu (Univ. Bordeaux)
Lieu :
Lien web :
Résumé : Let A and B be (finite) subsets of abelian groups, and t a positive integer. Call :A->B local (or Freiman's) t-homomorphism if for anya_1,...,a_t in A f(a_1+...+a_t)=f(a_1)+...f(a_t). Call f local t-isomorphism if it is bijective and the inverse map is a local t-homomorphism as well. We say that a set is t-rectifiable if it is locally t-isomorphic to a set of integers. The notions of local isomorphism and rectifiability are basic in the modern additive combinatorics. It is easy to see that a finite subset of a torsion-free abelian group is t-rectifiable for any t. Recently, Seva Lev determined the t-rectifiability threshold of an arbitrary abelian group. (The t-rectifiability threshold of G is such r=r_t(G) that any subset of r or less elements is rectifiable, but there exists a non-rectifiable subset of r+1 elements.) This remarkable work of Lev will be the subject of my talk. In the course, we shall also discuss Schinzel's beautiful upper bound for determinants.