Finding involutions in finite Lie type groups of odd characteristic

Let G be a finite group of Lie type in odd characteristic defined over a field with q elements. We prove that there is an absolute (and explicit) constant c such that, if G is a classical matrix group of dimension n⩾2, then at least c/log(n) of its elements are such that some power is an involution with fixed point subspace of dimension in the interval [n/3,2n/3). If G is exceptional, or G is classical of small dimension, then, for each conjugacy class C of involutions, we find a very good lower bound for the proportion of elements of G for which some power lies in C.