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 $\mathbb{F}_p$.

中文翻译：

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稳定规律的群版

我们证明，给定 ε > 0 并且ķ≥1，存在整数n使得以下成立。认为G是一个有限群并且一种⊆G是ķ-稳定的。那么有一个正规子群H≤G最多的索引n, 和一组是⊆G，它是陪集的并集H, 这样 |一种△是| ≤ε|H|。由此可见，对于任何陪集C的H, 要么 |C∩一种|≤ε|H| 或 |C\一种| ≤ ε |H|。这定性地概括了 Terry 和 Wolf 最近关于向量空间的工作$\mathbb{F}_p$.