VC维数有界的集合的卷积是uniformly continuous
Convolutions of sets with bounded VC-dimension are uniformly continuous
摘要：
我们研究了VC-维的概念的子集的群体，定义这一集$A$是VC-维的家庭$\{ (xA) \cap A : x \in A\cdot A^{-1} \}$。证明了如果阿贝尔群的有限子集$A$具有有界VC维数，则卷积$1_A*1_{-A}$在定量上是玻尔一致连续的.这推广和加强了一个版本的稳定算术正则引理特里和沃尔夫在各种方式。特别是，它直接意味着多项式Bogolyubov-Ruzsa猜想-一个强版本的多项式Freiman-Ruzsa猜想-持有有限VC-维的集合。我们还证明了一些结果在非阿贝尔设置。在某种意义上，这给出了具有VC-有界维数的平移闭集系统的结构定理：如果一个阿贝尔群的VC-有界子集族在平移下是闭的，则每个成员都有一个简单的玻尔集描述，直到一个小的误差。
英文摘要：
We study a notion of VC-dimension for subsets of groups, defining this for a set $A$ to be the VC-dimension of the family $\{ (xA) \cap A : x \in A\cdot A^{-1} \}$. We show that if a finite subset $A$ of an abelian group has bounded VC-dimension, then the convolution $1_A*1_{-A}$ is Bohr uniformly continuous, in a quantitatively strong sense. This generalises and strengthens a version of the stable arithmetic regularity lemma of Terry and Wolf in various ways. In particular, it directly implies that the Polynomial Bogolyubov--Ruzsa Conjecture -- a strong version of the Polynomial Freiman--Ruzsa Conjecture -- holds for sets with bounded VC-dimension. We also prove some results in the non-abelian setting. In some sense, this gives a structure theorem for translation-closed set systems with bounded (classical) VC-dimension: if a VC-bounded family of subsets of an abelian group is closed under translation, then each member has a simple description in terms of Bohr sets, up to a small error.
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