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A REFINED WARING PROBLEM FOR FINITE SIMPLE GROUPS
Published online by Cambridge University Press: 17 March 2015
Abstract
Let $w_{1}$ and
$w_{2}$ be nontrivial words in free groups
$F_{n_{1}}$ and
$F_{n_{2}}$, respectively. We prove that, for all sufficiently large finite nonabelian simple groups
$G$, there exist subsets
$C_{1}\subseteq w_{1}(G)$ and
$C_{2}\subseteq w_{2}(G)$ 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
$G$.
MSC classification
- Type
- Research Article
- Information
- Creative Commons
- This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/3.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
- Copyright
- © The Author(s) 2015
References
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