  Let v and w be nontrivial words in two free groups. We prove that, for all
sufficiently large finite non-abelian simple groups G, there exist subsets C of
v(G) and D of w(G) of size such that every element of G can be realized in at
least one way as the product of an element of C and an element of D and the
average number of such representations is O(log |G|). In particular, if w is a
fixed nontrivial word and G is a sufficiently large finite non-abelian simple
group, then w(G) contains a thin base of order 2. This is a non-abelian
analogue of a result of Van Vu for the classical Waring problem. Further
results concerning thin bases of G of order 2 are established for any finite
group and for any compact Lie group G.
