A Refined Waring Problem for Finite Simple Groups

Let w1 and w2 be nontrivial words in free groups Fn1 and Fn2 , respectively. We prove that, for all sufficiently large finite nonabelian simple groups G, there exist subsets C1 ⊆ w1(G) and C2 ⊆ w2(G) such that |Ci | = O(|G|1/2 log |G|) and C1C2 = G. In particular, ifw is any nontrivial word and G is a sufficiently large finite nonabelian simple group, then w(G) contains a thin base of order 2. This is a nonabelian analog of a result of Van Vu [‘On a refinement of Waring’s problem’, Duke Math. J. 105(1) (2000), 107–134.] 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. 2010 Mathematics Subject Classification: 20C33 (primary); 20D06 (secondary)