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A group version of stable regularity

Published online by Cambridge University Press:  24 October 2018

G. CONANT ,
A. PILLAY  and
C. TERRY
G. CONANT
Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, IN, U.S.A.
A. PILLAY
Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, IN, U.S.A.
C. TERRY
Affiliation:
Department of Mathematics, The University of Chicago, Chicago, IL, U.S.A. e-mail: [email protected]
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Abstract

We prove that, given ε > 0 and k ≥ 1, there is an integer n such that the following holds. Suppose G is a finite group and A ⊆ G is k-stable. Then there is a normal subgroup H ≤ G of index at most n, and a set Y ⊆ G, which is a union of cosets of H, such that |A △ Y| ≤ε|H|. It follows that, for any coset C of H, either |C ∩ A|≤ ε|H| or |C \ A| ≤ ε |H|. This qualitatively generalises recent work of Terry and Wolf on vector spaces over $\mathbb{F}_p$.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 168 , Issue 2 , March 2020 , pp. 405 - 413
Copyright
Copyright © Cambridge Philosophical Society 2018 

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Footnotes

†

Supported by NSF grants DMS-1360702 and DMS-1665035.

References

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