论文标题
任意有限组的稳定集的定量结构
Quantitative structure of stable sets in arbitrary finite groups
论文作者
论文摘要
我们表明，有限组中的$ k $稳定设置可以近似，直到给出的错误$ε> 0 $，索引的索引子组的左cosets $ε^{\ text { - } o_k（1）} $。这改善了Terry和Wolf在有限的Abelian群体中稳定算术规律性的类似结果，并导致对作者，Pillay和Terry在任意有限群体中的稳定套装的作品进行定量描述。我们还证明了有限稳定的小组中有限稳定的小组的相似结果，该组提供了Martin-Pizarro，Palacín和Wolf最近作品的定量版本。我们的证明将结果用于VC维度，以及稳定组理论的模型理论技术的有限化。
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.
