Given an Abelian group G, a Boolean-valued function f: G -> {-1,+1}, is said to be s-sparse, if it has at most s-many non-zero Fourier coefficients over the domain G. In a seminal paper, Gopalan et al. proved "Granularity" for Fourier coefficients of Boolean valued functions over Z_2^n, that have found many diverse applications in theoretical computer science and combinatorics. They also studied structural results for Boolean functions over Z_2^n which are approximately Fourier-sparse. In this work, we obtain structural results for approximately Fourier-sparse Boolean valued functions over Abelian groups G of the form,G:= Z_{p_1}^{n_1} \times ... \times Z_{p_t}^{n_t}, for distinct primes p_i. We also obtain a lower bound of the form 1/(m^{2}s)^ceiling(phi(m)/2), on the absolute value of the smallest non-zero Fourier coefficient of an s-sparse function, where m=p_1 ... p_t, and phi(m)=(p_1-1) ... (p_t-1). We carefully apply probabilistic techniques from Gopalan et al., to obtain our structural results, and use some non-trivial results from algebraic number theory to get the lower bound. We construct a family of at most s-sparse Boolean functions over Z_p^n, where p > 2, for arbitrarily large enough s, where the minimum non-zero Fourier coefficient is 1/omega(n). The "Granularity" result of Gopalan et al. implies that the absolute values of non-zero Fourier coefficients of any s-sparse Boolean valued function over Z_2^n are 1/O(s). So, our result shows that one cannot expect such a lower bound for general Abelian groups. Using our new structural results on the Fourier coefficients of sparse functions, we design an efficient testing algorithm for Fourier-sparse Boolean functions, thata requires poly((ms)^phi(m),1/epsilon)-many queries. Further, we prove an Omega(sqrt{s}) lower bound on the query complexity of any adaptive sparsity testing algorithm.

翻译：给定一个阿贝尔群 G，若布尔值函数 f: G -> {-1,+1} 在定义域 G 上至多具有 s 个非零傅里叶系数，则称其为 s-稀疏的。在一篇开创性论文中，Gopalan 等人证明了 Z_2^n 上布尔值函数的傅里叶系数具有"粒度性"，该结论在理论计算机科学和组合数学中得到了广泛的应用。他们还研究了 Z_2^n 上近似傅里叶稀疏布尔函数的结构性结果。在本工作中，我们针对形式为 G:= Z_{p_1}^{n_1} \times ... \times Z_{p_t}^{n_t}（其中 p_i 为互异素数）的阿贝尔群 G，获得了近似傅里叶稀疏布尔值函数的结构性结果。我们还推导出 s-稀疏函数的最小非零傅里叶系数绝对值的一个下界，其形式为 1/(m^{2}s)^ceiling(phi(m)/2)，其中 m=p_1 ... p_t，phi(m)=(p_1-1) ... (p_t-1)。我们细致地应用了 Gopalan 等人论文中的概率技术以获得结构性结果，并借助代数数论中的一些非平凡结论来得到下界。我们构造了 Z_p^n（其中 p > 2）上的一族至多 s-稀疏布尔函数（s 可任意充分大），其最小非零傅里叶系数为 1/omega(n)。Gopalan 等人的"粒度性"结果意味着 Z_2^n 上任何 s-稀疏布尔值函数的非零傅里叶系数绝对值为 1/O(s)。因此，我们的结果表明对于一般阿贝尔群不能期望存在这样的下界。利用我们关于稀疏函数傅里叶系数的新结构性结果，我们设计了一种高效的傅里叶稀疏布尔函数测试算法，该算法需要 poly((ms)^phi(m),1/epsilon) 次查询。此外，我们证明了任何自适应稀疏性测试算法的查询复杂度存在 Omega(sqrt{s}) 的下界。