Tame regularity theorems for groups with a distinguished subset

(This is joint with  Gabriel Conant and Caroline Terry.)
Graph regularity theorems (i.e. Szemeredi) concern decomposing finite graphs
(V,W,R) into a small number of subgraphs (Vi,Wj,R|(V_i×W_j)) most of
which are  "almost regular", i.e. subgraphs have approximately the same
density.
When more assumptions are made on the relation R such as uniform stability
or NIP one obtains stronger statements with almost homogeneity in place of
almost regularity.
In the group version we consider finite groups G equipped with a
distinguished subset A and assumptions are made on the relation xy  A. One
seeks nice decompositions compatible with the group structure and this is
what I will talk about.