### Abstract

Let w_{1} and w_{2} be nontrivial words in free groups F_{n1} and F_{n2}, respectively. We prove that, for all sufficiently large finite nonabelian simple groups G, there exist subsets C_{1} ⊆ w_{1}(G) and C_{2} ⊆ w_{2}(G) such that |C_{i}| = O(|G|^{1/2} log^{1/2} |G|) and C_{1}C_{2} = G. In particular, if w 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.

Original language | English (US) |
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Article number | e6 |

Journal | Forum of Mathematics, Sigma |

Volume | 3 |

DOIs | |

State | Published - Jan 1 2015 |

### ASJC Scopus subject areas

- Algebra and Number Theory
- Analysis
- Computational Mathematics
- Discrete Mathematics and Combinatorics
- Geometry and Topology
- Mathematical Physics
- Statistics and Probability
- Theoretical Computer Science

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## Cite this

*Forum of Mathematics, Sigma*,

*3*, [e6]. https://doi.org/10.1017/fms.2015.4