On the minimal fourier degree of symmetric Boolean functions

In this paper we give a new upper bound on the minimal degree of a nonzero Fourier coefficient in any non-linear symmetric Boolean function. Specifically, we prove that for every non-linear and symmetric f: {0, 1}^k → {0, 1} there exists a set (Formula presented) such that {pipe}S{pipe} = O(Γ(k)+√k, and f(S)̂ ≠ 0 where Γ(m)≤m ^0.525 is the largest gap between consecutive prime numbers in {1,..., m}. As an application we obtain a new analysis of the PAC learning algorithm for symmetric juntas, under the uniform distribution, of Mossel et al. [10]. Our bound on the degree is a significant improvement over the previous result of Kolountzakis et al. [8] who proved that {pipe}S{pipe}=O(k=log k). We also show a connection between lower-bounding the degree of non-constant functions that take values in {0,1,2} and the question that we study here.