任意有限群中稳定集的数量结构
Quantitative structure of stable sets in arbitrary finite groups
摘要：
我们证明了有限群中的$k$独立集在给定误差$ε>0$的情况下可以用指数为$ε^{\text{-}O_k(1)}$的子群的左陪集来近似.这提高了限制在一个类似的结果特里和沃尔夫稳定算术正则性有限阿贝尔群，并导致定量帐户的工作的作者，皮莱和特里稳定集在任意有限群。我们还证明了一个类似的结果，有限稳定集的小三倍在任意群，这提供了一个定量的版本，最近的工作由马丁-皮萨罗，Palacín和沃尔夫。我们的证明使用VC维的结果，和有限化的模型理论技术从稳定的群论。
英文摘要：
We show that a $k$-stable set in a finite group can be approximated, up to given error $ε>0$, by left cosets of a subgroup of index $ε^{\text{-}O_k(1)}$. 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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