Linear Approximate Groups

This is an informal announcement of results to be described and proved in detail in a paper to appear. We give various results on the structure of approximate subgroups in linear groups such as ${\text{SL}}_n(k)$. For example, generalising a result of Helfgott (who handled the cases $n=2$ and 3), we show that any approximate subgroup of ${\text{SL}}_n({\text{F}}_q)$ which generates the group must be either very small or else nearly all of ${\text{SL}}_n({\text{F}}_q)$. The argument is valid for all Chevalley groups $G({\text{F}}_q)$.