A Refined Waring Problem for Finite Simple Groups

TL;DR

This paper investigates the Waring problem in finite simple groups, showing that large groups can be covered efficiently by products of subsets derived from nontrivial words, extending classical results to non-abelian groups.

Contribution

It proves the existence of thin bases of order 2 in large finite simple groups and extends the Waring problem to non-abelian groups and compact Lie groups.

Findings

Existence of subsets covering all elements with O(log |G|) representations

Large finite simple groups contain thin bases of order 2

Results extend to finite and compact Lie groups

Abstract

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.