Convolutions of sets with bounded VC-dimension are uniformly continuous
Convolutions of sets with bounded VC-dimension are uniformly continuous
复制标题
具有有界 VC 维数的集合的卷积是一致连续的
DOI：
10.19086/da.18561
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发表时间：
2018
期刊：
影响因子：
--
通讯作者：
Olof Sisask
中科院分区：
文献类型：
--
作者：
Olof Sisask
We introduce a notion of VC-dimension for subsets of groups, defining this for a set $A$ to be the VC-dimension of the family $\{ A \cap(xA) : x \in A\cdot A^{-1} \}$. 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.
