  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 $\SL_n(k)$. For example,
generalising a result of Helfgott (who handled the cases $n = 2$ and 3), we
show that any approximate subgroup of $\SL_n(\F_q)$ which generates the group
must be either very small or else nearly all of $\SL_n(\F_q)$. The argument is
valid for all Chevalley groups $G(\F_q)$.
