A Sauer-Shelah-Perles lemma for sumsets
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].
(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].
We show that any family of subsets satisfies $\lvert A\rvert \leq O\bigl(n^{\lceil{d}/{2}\rceil}\bigr)$, where is the VC dimension of , and is the symmetric difference operator. We also observe that replacing by either or fails to satisfy an analogous statement. Our proof is based on the polynomial method; specifically, on an argument due to [Croot, Lev, Pach '17].
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