Group Action Combinatorics

This paper generalizes the basic notions of additive and multiplicative combinatorics to the setting of group actions: if $G$ is a group acting on a set $X$, and we have subsets $A\subseteq G$ and $Y\subseteq X$ such that the set of pairs $g\cdot y$ with $g\in A,y\in Y$ is not much larger than $Y$, what structure must $A$ and $Y$ have? Briefly, what is the structure of sets with small "image set"? In this setting, we develop analogs of Ruzsa's triangle inequality, covering theorems, multiplicative energy, and the Balog-Szemer\'{e}di-Gowers theorem. Approximate stabilizers, which we call symmetry sets, play an important role. While our focus is on presenting a general theory, we answer the inverse image set question in some special cases. To do so, we combine the group action version of the Balog-Szemer\'{e}di-Gowers theorem with structure theorems for approximate groups and bounds for the sizes of symmetry sets.