Testing Sumsets is Hard

A subset S of the Boolean hypercube 𝔽_2^n is a *sumset* if S = {a + b : a, b∈ A} for some A ⊆𝔽_2^n. Sumsets are central objects of study in additive combinatorics, featuring in several influential results. We prove a lower bound of Ω(2^n/2) for the number of queries needed to test whether a Boolean function f:𝔽_2^n →{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)·poly(n)-query algorithm for a smoothed analysis formulation of the sumset *refutation* problem.