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arXiv · 2401.07242

Testing Sumsets is Hard

Xi Chen·Shivam Nadimpalli·Tim Randolph·Rocco A. Servedio·Or Zamir
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Abstract

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.

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Xi Chen, Shivam Nadimpalli, Tim Randolph, Rocco A. Servedio, Or Zamir. 2024-01-14. Testing Sumsets is Hard. https://arxiv.org/abs/2401.07242

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