Growth in linear groups

We prove a conjecture of Helfgott on the structure of sets of bounded tripling in bounded rank, which states the following. Let $A$ be a finite symmetric subset of ${\mathrm{G}\mathrm{L}}_n(\mathbf{F})$ for any field $\mathbf{F}$ such that $\mathrm{\mid }{A}^3\mathrm{\mid }\le K\mathrm{\mid }A\mathrm{\mid }$. Then there are subgroups $H⊴\mathrm{\Gamma }⊴⟨A⟩$ such that $A$ is covered by $K^{O_n(1)}$ cosets of $\mathrm{\Gamma }$, $\mathrm{\Gamma }\mathrm{/}H$ is nilpotent of step at most $n-1$, and $H$ is contained in $A^{O_n(1)}$. This theorem includes the Product Theorem for finite simple groups of bounded rank as a special case. As an application of our methods we also show that the diameter of sufficiently quasirandom finite linear groups is poly-logarithmic.