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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

REFERENCES

[1] Green, B. A Szemerédi-type regularity lemma in abelian groups, with applications. Geom. Funct. Anal. 15 (2005), no. 2, 340–376.Google Scholar
[2] Hrushovski, E. Pseudo-finite fields and related structures, Model theory and applications. Quad. Mat. vol. 11 (Aracne, Rome, 2002), pp. 151–212.Google Scholar
[3] Hrushovski, E. Stable group theory and approximate subgroups. J. Amer. Math. Soc. 25 (2012), no. 1, 189–243.Google Scholar
[4] Hrushovski, E., Peterzil, Y. and Pillay, A. Groups, measures, and the NIP. J. Amer. Math. Soc. 21 (2008), no. 2, 563–596.Google Scholar
[5] Hrushovski, E. and Pillay, A. Groups definable in local fields and pseudo-finite fields. Israel J. Math. 85 (1994), no. 1-3, 203–262.Google Scholar
[6] Malliaris, M. and Pillay, A. The stable regularity lemma revisited. Proc. Amer. Math. Soc. 144 (2016), no. 4, 1761–1765.Google Scholar
[7] Malliaris, M. and Shelah, S. Regularity lemmas for stable graphs. Trans. Amer. Math. Soc. 366 (2014), no. 3, 1551–1585.Google Scholar
[8] Newelski, L. and Petrykowski, M. Weak generic types and coverings of groups. I. Fund. Math. 191 (2006), no. 3, 201–225.Google Scholar
[9] Pillay, A. Geometric stability theory. Oxford Logic Guides, vol. 32. (The Clarendon Press, Oxford University Press, New York, 1996, Oxford Science Publications).Google Scholar
[10] Shelah, S. Classification theory and the number of nonisomorphic models, second ed. Studies in Logic and the Foundations of Mathematics, vol. 92 (North-Holland Publishing Co., Amsterdam, 1990).Google Scholar
[11] Terry, C. and Wolf, J.. Stable arithmetic regularity in the finite-field model. arXiv:1710.02021 (2017).Google Scholar