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; $\not 0 \ne S \subset [k]$ such that ¦S¦ = O(Γ(k)+√k, and $\hat f(S) \ne 0$ where Γ(m)≤m0.525 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 ¦S¦=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.