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 {$2^{O(n/2)} \cdot \mathrm{poly}(n)$}-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（sumset），如果存在 \(A \subseteq \mathbb{F}_2^n\) 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，出现在多个重要成果中。我们证明，测试布尔函数 \(f:\mathbb{F}_2^n \to \{0,1\}\) 是否为和集的指示函数所需的查询次数下界为 \(\Omega(2^{n/2})\)。这一测试和集的下界源于对相关“移位测试”（shift testing）问题的尖锐界，该问题本身可能具有独立的研究价值。此外，针对和集的反驳（refutation）问题的平滑分析形式，我们给出了一个近乎最优的算法，其查询复杂度为 \(2^{O(n/2)} \cdot \mathrm{poly}(n)\)。