A REFINED WARING PROBLEM FOR FINITE SIMPLE GROUPS

Michael Larsen; Pham Huu Tiep

doi:10.1017/fms.2015.4

A REFINED WARING PROBLEM FOR FINITE SIMPLE GROUPS

Michael Larsen, Pham Huu Tiep

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

Let w₁ and w₂ 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 C₁ ⊆ w₁(G) and C₂ ⊆ w₂(G) such that |C_i| = O(|G|^1/2 log^1/2 |G|) and C₁C₂ = 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)
Article number	e6
Journal	Forum of Mathematics, Sigma
Volume	3
DOIs	https://doi.org/10.1017/fms.2015.4
State	Published - Jan 1 2015
Externally published	Yes

ASJC Scopus subject areas

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

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title = "A REFINED WARING PROBLEM FOR FINITE SIMPLE GROUPS",

abstract = "Let w1 and w2 be nontrivial words in free groups Fn1 and Fn2, respectively. We prove that, for all sufficiently large finite nonabelian simple groups G, there exist subsets C1 ⊆ w1(G) and C2 ⊆ w2(G) such that |Ci| = O(|G|1/2 log1/2 |G|) and C1C2 = 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.",

author = "Michael Larsen and Tiep, {Pham Huu}",

note = "Funding Information: Michael Larsen was partially supported by NSF grant DMS-1101424 and a Simons Fellowship. Pham Huu Tiep was partially supported by NSF grant DMS- 1201374 and the Simons Foundation Fellowship 305247. Publisher Copyright: {\textcopyright} 2015 The Author(s).",

year = "2015",

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doi = "10.1017/fms.2015.4",

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T1 - A REFINED WARING PROBLEM FOR FINITE SIMPLE GROUPS

AU - Larsen, Michael

AU - Tiep, Pham Huu

N1 - Funding Information: Michael Larsen was partially supported by NSF grant DMS-1101424 and a Simons Fellowship. Pham Huu Tiep was partially supported by NSF grant DMS- 1201374 and the Simons Foundation Fellowship 305247. Publisher Copyright: © 2015 The Author(s).

PY - 2015/1/1

Y1 - 2015/1/1

N2 - Let w1 and w2 be nontrivial words in free groups Fn1 and Fn2, respectively. We prove that, for all sufficiently large finite nonabelian simple groups G, there exist subsets C1 ⊆ w1(G) and C2 ⊆ w2(G) such that |Ci| = O(|G|1/2 log1/2 |G|) and C1C2 = 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.

AB - Let w1 and w2 be nontrivial words in free groups Fn1 and Fn2, respectively. We prove that, for all sufficiently large finite nonabelian simple groups G, there exist subsets C1 ⊆ w1(G) and C2 ⊆ w2(G) such that |Ci| = O(|G|1/2 log1/2 |G|) and C1C2 = 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.

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JO - Forum of Mathematics, Sigma

JF - Forum of Mathematics, Sigma

M1 - e6

ER -

A REFINED WARING PROBLEM FOR FINITE SIMPLE GROUPS

Abstract

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