We prove Bogolyubov–Ruzsa-type results for finite subsets of groups with small tripling, |A3| ≤ O(|A|), or small alternation, |AA−1A| ≤ O(|A|). As applications, we obtain a qualitative analogue of Bogolyubov’s lemma for dense sets in arbitrary finite groups, as well as a quantitative arithmetic regularity lemma for sets of bounded VC-dimension in finite groups of bounded exponent. The latter result generalizes the abelian case, due to Alon, Fox and Zhao, and gives a quantitative version of previous work of the author, Pillay and Terry.

中文翻译：

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关于任意组中的小三倍或小交替的有限集

我们证明了具有小三倍组的有限子集的 Bogolyubov-Ruzsa 型结果，|一种3| ≤○(|一种|)，或小的交替，|AA-1一种| ≤○(|一种|)。作为应用，我们获得了任意有限群中稠密集的 Bogolyubov 引理的定性模拟，以及有界指数有限群中的有界 VC 维集的定量算术正则性引理。后一个结果概括了 Alon、Fox 和 Zhao 的阿贝尔案例，并给出了作者 Pillay 和 Terry 先前工作的定量版本。