A subset $S$ of the Boolean hypercube $\mathbb{F}_2^n$ is a sumset if $S = A+A = \{a + b \ | \ a, b\in A\}$ for some $A \subseteq \mathbb{F}_2^n$. We prove that the number of sumsets in $\mathbb{F}_2^n$ is asymptotically $(2^n-1)2^{2^{n-1}}$. Furthermore, we show that the family of sumsets in $\mathbb{F}_2^n$ is almost identical to the family of all subsets of $\mathbb{F}_2^n$ that contain a complete linear subspace of co-dimension $1$.

翻译：若布尔超立方体 $\mathbb{F}_2^n$ 的子集 $S$ 满足 $S = A+A = \{a + b \ | \ a, b\in A\}$ 对于某个 $A \subseteq \mathbb{F}_2^n$，则称 $S$ 为和集。我们证明 $\mathbb{F}_2^n$ 中和集的数量渐近等于 $(2^n-1)2^{2^{n-1}}$。此外，我们证明 $\mathbb{F}_2^n$ 中的和集族几乎等同于包含一个余维数为 $1$ 的完全线性子空间的所有子集族。