A REFINED WARING PROBLEM FOR FINITE SIMPLE GROUPS
Forum of Mathematics, Sigma, Tome 3 (2015)
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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 $G$.
@article{10_1017_fms_2015_4,
author = {MICHAEL LARSEN and PHAM HUU TIEP},
title = {A {REFINED} {WARING} {PROBLEM} {FOR} {FINITE} {SIMPLE} {GROUPS}},
journal = {Forum of Mathematics, Sigma},
publisher = {mathdoc},
volume = {3},
year = {2015},
doi = {10.1017/fms.2015.4},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2015.4/}
}
MICHAEL LARSEN; PHAM HUU TIEP. A REFINED WARING PROBLEM FOR FINITE SIMPLE GROUPS. Forum of Mathematics, Sigma, Tome 3 (2015). doi: 10.1017/fms.2015.4
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