On Fourier analysis of sparse Boolean functions over certain Abelian groups

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 Z2^n, that have found many diverse applications in theoretical computer science and combinatorics. They also studied structural results for Boolean functions over Z2ⁿ 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{p1}^{n_1} \times ... \times Z{pt}^{n_t}, for distinct primes pi. 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=p1 ... pt, and phi(m)=(p1-1) ... (pt-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 Zp^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_2ⁿ 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.