Structure of sparse Boolean functions over Abelian groups, and its application to testing

TL;DR

This work extends the Fourier-sparsity analysis of Boolean functions from the hypercube to arbitrary finite Abelian groups by introducing a generalized granularity notion relative to the group exponent $Δ$ and proving two key structure theorems: (i) large Fourier coefficients of an $μ$-close-to-$s$-sparse function are near $k$-granular with $k=⌈\log_Δ(2s)⌉$, and (ii) there exists a close $s$-sparse Boolean function when $μ≤1/(8·2^{φ(Δ)}s^{φ(Δ)})$. It also provides a Δ-dependent lower bound $|\hat f(χ)|≥1/2^{φ(Δ)/2} s^{φ(Δ)/2}$ for coefficients in the support and a Δ-independent bound, supported by algebraic-number-theory techniques. Building on these structural results, the authors design both nonadaptive and adaptive sparsity testers with tight-ish complexity: nonadaptive query complexity poly$((2s)^{φ(Δ)},1/ε)$ and adaptive lower bound $Ω(√s)$. They further construct Boolean examples with unusually small Fourier coefficients for $p≥5$, discuss the degree over $G$, and establish lower bounds via random coset projections and spectral bucketing. Together these results generalize the foundational sparsity-testing framework of Gopalan et al. to general Abelian groups, with implications for sparse Fourier transforms and learning in broader algebraic domains.

Abstract

We study Fourier-sparse Boolean functions over general finite Abelian groups. A Boolean function $f : G \to \{-1,+1\}$ is $s$-sparse if it has at most $s$ non-zero Fourier coefficients. We introduce a general notion of granularity of Fourier coefficients and prove that every non-zero coefficient of an $s$-sparse Boolean function has magnitude at least \begin{equation*} \frac{1}{2^{\varphi(Δ)/2} \, s^{\varphi(Δ)/2}}, \end{equation*} where $Δ$ denotes the exponent of the group $G$ (that is, the maximum order of an element in $G$) and $\varphi$ is the Euler's totient function. This generalizes the celebrated result of Gopalan et al. (SICOMP 2011) for $\mathbb{Z}_2^n$, extending it to all finite Abelian groups via new techniques from group theory and algebraic number theory. Using our new structural results on the Fourier coefficients of sparse functions, we design an efficient sparsity testing algorithm for Boolean functions. The tester distinguishes whether a given function is $s$-sparse or $ε$-far from every $s$-sparse Boolean function, with query complexity $poly\left((2s)^{\varphi(Δ)},1/ε\right)$. In addition, we generalize the classical notion of Boolean degree to arbitrary Abelian groups and establish an $Ω(\sqrt{s})$ lower bound for adaptive sparsity testing.

Theorems & Definitions (120)

• Definition 1.1
• Definition 1.2
• Theorem 1.3: Structure theorem 1
• Theorem 1.4: Structure theorem 2
• Theorem 1.5: Fourier coefficient lower bound
• Remark
• Theorem 1.6: Small Fourier coefficients
• Definition 1.7
• Theorem 1.8: Testing $s$-sparsity
• Theorem 1.9