Abstract

The structure of finite subsets A of an ambient algebraic group G, which do not grow much under multiplication, say |AA|<|A|^{, is well understood after the works of Hrushovski, Pyber-Szabo and Breuillard-Green-Tao on approximate subgroups of algebraic groups. A more general question, tackled by Elekes and Szabo, asks for the structure of Cartesian products A_1 \times … \times A_n of finite subsets of size N of an arbitrary d-dimensional algebraic variety W, with large (i.e. >N}{\dim V/d}) intersection with a given subvariety V \leq W^n (the case n=3, W=G, A_i=A, V={(x,y,xy)} corresponds to the above mentioned approximate group problem). In joint work with Martin Bays, we completely characterize the algebraic varieties V that can admit a (general position) family of such finite Cartesian products with large intersection. We show that they are in algebraic correspondence with a subgroup of a commutative algebraic group endowed with an extra structure arising from a certain division ring of group endomorphisms. The proof makes use of the Veblen-Young theorem on abstract projective geometries, generalized Szemeredi-Trotter bounds and Hrushovski’s formalism of pseudo-finite dimensions.