Convolutions of sets with bounded VC-dimension are uniformly continuous

  title={Convolutions of sets with bounded VC-dimension are uniformly continuous},
  author={Olof Sisask},
  journal={arXiv: Combinatorics},
  • Olof Sisask
  • Published 8 February 2018
  • Mathematics
  • arXiv: Combinatorics
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… 

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