2013/12/17 by Larsen, Michael, Tiep, Pham Huu
#11P05 #20C33 #20D06 (Primary) 11B13 #22C05 (Secondary) #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.1312.4998
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.