Abstract:We show that a $k$-stable set in a finite group can be approximated, up to given error $\epsilon >0$, by left cosets of a subgroup of index $\epsilon ^{\text {-} O_k(1)}$. This improves the bound in a similar result of Terry and Wolf on stable arithmetic regularity in finite abelian groups, and leads to a quantitative account of work of the author, Pillay, and Terry on stable sets in arbitrary finite groups. We also prove an analogous result for finite stable sets of small tripling in arbitrary groups, which provides a quantitative version of recent work by Martin-Pizarro, Palacín, and Wolf. Our proofs use results on VC-dimension, and a finitization of model-theoretic techniques from stable group theory.

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中文翻译：

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任意有限群中稳定集的定量结构

摘要：我们表明，在给定误差 $\epsilon >0$ 的情况下，有限群中的 $k$-stable 集可以通过索引 $\epsilon ^{\text {-} O_k 的子群的左陪集来逼近(1)}$。这改进了 Terry 和 Wolf 在有限阿贝尔群中稳定算术正则性的类似结果中的界限，并导致对作者、Pillay 和 Terry 对任意有限群中稳定集的工作进行定量描述。我们还证明了任意群中小三重的有限稳定集的类似结果，它提供了 Martin-Pizarro、Palacín 和 Wolf 最近工作的定量版本。我们的证明使用了 VC 维度的结果，以及来自稳定群论的模型理论技术的有限化。

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