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).Comment: 11 pages. Submitted, Electronic Research Announcements. Small change