The structure of a function and small images

Abstract: It is well known that if $G$ is an abelian group and $A$ is a finite subset of $G$ which minimises the size of $A+A$ for a fixed size of $A$, then $A$ must essentially be as close as possible to being a coset of some subgroup of $G$: for instance, if $G$ is finite then the set $A$ must be a coset in $G$ (provided that $G$ has subgroups with size $|A|$), and if $G=\mathbb{Z}$ then the set $A$ must be an arithmetic progression. One may wish to describe an extension of this phenomenon to more arbitrary functions than addition on abelian groups: if $X$ is a set and $F: X \times X \rightarrow X$ is a function, then can the structure of $F$ (to the extent that there is some) be read from the structure of the sets $A$ minimising the size of the image $F(A,A)$ for a fixed size of $A$ ? We will discuss the progress that has been made in this direction and what remains to be understood.

analysis of PDEsclassical analysis and ODEsfunctional analysis

Audience: researchers in the topic

OARS Online Analysis Research Seminar

Series comments: Visit our homepage for further information. Some recordings available on YouTube