Structure and regularity for subsets of groups with finite VC-dimension

Suppose $G$ is a finite group and $A\\subseteq G$ such that ${gA:g\\in G}$ has VC-dimension strictly less than $k$. We find algebraically well-structured sets in which, up to chosen $\\epsilon>0$, describe the structure of $A$ behave regularly with respect translates $A$. For subclass groups uniformly fixed exponent $r$, these algebraic objects are normal subgroups index bounded terms $k$, $\\epsilon$. arbitrary groups, we use Bohr neighborhoods rank width inside index. Our proofs largely model-theoretic, heavily rely on structural analysis compactifications pseudofinite as inverse limits Lie groups. The introduction into nonabelian setting uses model-theoretic methods related work Breuillard, Green, Tao \[8] Hrushovski \[28] approximate well result Alekseev, Glebskiĭ, Gordon \[1] homomorphisms.