SEMINARIO TEORÍA DE NÚMEROS

Title: An inverse theorem for Freiman multi-homomorphisms and its applications

SPEAKER: Luka Milićević (Mathematical Institute of the Serbian Academy of Sciences and Arts)

DATE: Monday, June 13th - 17:30

PLACE: Online, Microsoft Teams (código: owfo832)

ABSTRACT: In the field of additive combinatorics, one is frequently interested in approximate versions of algebraic structures. One of the key examples of such objects is a Freiman homomorphism. This is a map Phi defined on a subset A of an abelian group G mapping its elements to another abelian group H with the property that whenever a,b,c,d in A satisfy a + b = c + d then Phi(a) + Phi(b) = Phi(c) + Phi(d). When Gand H are vector spaces over a prime field F_p and A is sufficiently
dense, it turns out that Freiman homomorphisms essentially come from restrictions of affine maps (which satisfy the same property, but are defined on whole group).

Let now G_1,..., G_k be vector spaces over F_p. In this talk I am interested in a multidimensional generalization of the notion of a Freiman homomorphism. We say that a map Phi defined on a subset of the product G_1 x ... x G_k is a Freiman multi-homomorphism if Phi is a Freiman homomorphism in every principal direction (i.e. when x_i in G_i is fixed for each i except one direction d, the map that sends
element x_d to Phi(x_1,..., x_k) is a Freiman homomorphism, where we allow those x_d for which (x_1,..., x_k) is in the domain of Phi).

It turns out that a Freiman multi-homorphism defined on a dense subset of G_1 x ... x G_k necessarily coincides with a global multiaffine map at many points. In this talk I will discuss the proof of this fact which Tim Gowers and I proved in a joint work. I will also discuss applications of this theorem and some related more recent developments.