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.

翻译：我们研究一般有限阿贝尔群上的傅里叶稀疏布尔函数。若布尔函数 $f : G \to \{-1,+1\}$ 至多具有 $s$ 个非零傅里叶系数，则称其为 $s$-稀疏函数。我们引入了傅里叶系数粒度的一般概念，并证明任意 $s$-稀疏布尔函数的每个非零系数幅值至少为 \begin{equation*} \frac{1}{2^{\varphi(Δ)/2} \, s^{\varphi(Δ)/2}}, \end{equation*} 其中 $Δ$ 表示群 $G$ 的指数（即 $G$ 中元素的最大阶），$\varphi$ 为欧拉函数。该结论推广了 Gopalan 等人（SICOMP 2011）在 $\mathbb{Z}_2^n$ 上的经典结果，通过群论与代数数论的新技术将其扩展至所有有限阿贝尔群。基于稀疏函数傅里叶系数的新结构结果，我们设计了一种高效的布尔函数稀疏性测试算法。该测试器能以 $poly\left((2s)^{\varphi(Δ)},1/ε\right)$ 的查询复杂度，区分给定函数是 $s$-稀疏函数还是与所有 $s$-稀疏布尔函数 $ε$-远离。此外，我们将经典的布尔次数概念推广至任意阿贝尔群，并为自适应稀疏性测试建立了 $Ω(\sqrt{s})$ 的下界。