Testing Sumsets is Hard

TL;DR

This work investigates the algorithmic recognition of sumsets in the Boolean cube $\mathbb{F}_2^n$, introducing sumset testing, shift testing, and smoothed sumset refutation. It proves a tight $\Theta(2^{n/2})$-level boundary for shift testing and, via a reduction, a matching $\Omega(2^{n/2})$ lower bound for sumset testing, while also delivering a near-optimal smoothed-refutation algorithm whose $0$-certificate size scales as $2^{n/2}\cdot\mathrm{poly}(n)/\varepsilon^{1.5}$. The results bridge additive combinatorics and sublinear property testing, revealing a precise complexity separation between shift and sumset testing and outlining pathways for strengthening refutation without smoothed assumptions. Open questions include removing the smoothed analysis constraint, closing the gap for sumset testing, and exploring intermediate problems like $k$-shift testing as a bridge between these tasks.

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 $Ω(2^{n/2})$ 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.

Figures (1)

• Figure 1: The set $\mathcal{S}(A, B) \subseteq \mathbb{F}_2^{n+2}$. By \ref{['prop:sumset-yes']}, for a typical $(\boldsymbol{A}_{\mathrm{yes}},\boldsymbol{B}_{\mathrm{yes}})$ drawn from $\mathcal{D}_{\mathrm{yes}}$, adding a single point $(1,1,\boldsymbol{s})$ in the top right cell makes $S(\boldsymbol{A}_{\mathrm{yes}},\boldsymbol{B}_{\mathrm{yes}})$ into a sumset.

Theorems & Definitions (41)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Definition 6
• Definition 7
• Claim 8
• proof
• Theorem 9
• Proposition 10