  We show that any family of subsets $A\subseteq 2^{[n]}$ satisfies $\lvert
A\rvert \leq O\bigl(n^{\lceil{d}/{2}\rceil}\bigr)$, where $d$ is the VC
dimension of $\{S\triangle T \,\vert\, S,T\in A\}$, and $\triangle$ is the
symmetric difference operator. We also observe that replacing $\triangle$ by
either $\cup$ or $\cap$ fails to satisfy an analogous statement. Our proof is
based on the polynomial method; specifically, on an argument due to [Croot,
Lev, Pach '17].
