Structure theorems for "approximate groups" show what structure remains when we relax the closure condition for groups. Examples of approximate groups are arithmetic progressions and large subsets of finite groups; a structure theorem shows how an arbitrary approximate group is composed of such examples. We show that if a subset A of GLn(q) is "approximately closed" under multiplication and the group G it generates is soluble, then there are subgroups U and S of G such that: A quickly generates U, S contains most of A, S/U is nilpotent. Briefly: approximate soluble linear groups are (almost) finite by nilpotent. This confirms a conjecture of Helfgott.
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