A subset $S$ of the Boolean hypercube $\mathbb{F}_2^n$ is a sumset if $S = \{a + b : a, b\in A\}$ for some $A \subseteq \mathbb{F}_2^n$. Sumsets are central objects of study in additive combinatorics, featuring in several influential results. We prove a lower bound of $\Omega(2^{n/2})$ for the number of queries needed to test whether a Boolean function $f:\mathbb{F}_2^n \to \{0,1\}$ is the indicator function of a sumset. Our lower bound for testing sumsets follows from sharp bounds on the related problem of shift testing, which may be of independent interest. We also give a near-optimal $2^{n/2} \cdot \mathrm{poly}(n)$-query algorithm for a smoothed analysis formulation of the sumset refutation problem.

翻译：布尔超立方体 $\mathbb{F}_2^n$ 的子集 $S$ 称为和集，若存在 $A \subseteq \mathbb{F}_2^n$ 使得 $S = \{a + b : a, b\in A\}$。和集是加性组合学中的核心研究对象，出现在多个重要结论中。我们证明了测试布尔函数 $f:\mathbb{F}_2^n \to \{0,1\}$ 是否为某个和集的指示函数所需查询次数的一个下界为 $\Omega(2^{n/2})$。这一测试和集的下界源于对相关移位测试问题的紧致界，该问题可能具有独立的研究价值。我们还针对和集反驳问题的平滑分析形式，给出了一个近乎最优的 $2^{n/2} \cdot \mathrm{poly}(n)$ 查询算法。