A Sauer-Shelah-Perles Lemma for Sumsets.

We show that any family of subsets A subset of 2([n]) satisfies vertical bar A vertical bar <= O(n([d/2])), where d is the VC dimension of {S Delta T vertical bar S, T is an element of A}, and Delta is the symmetric difference operator. We also observe that replacing Delta by either boolean OR or boolean AND fails to satisfy an analogous statement. Our proof is based on the polynomial method; specifically, on an argument due to [Croot, Lev, Pach '17].