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 Ø ≠ S ⊂ [k] such that |S| = O(Γ(k) + √k), and f̂(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. [JCSS, 2004]. Namely, we show that the running time of their algorithm is at most n^O(k0.525) · poly(n · 2^k,log · (1/δ)) where n is the number of variables, k is the size of the junta (i.e. number of relevant variables) and δ is the error probability. In particular, for k ≥ log(n)^{1/ (1-0-525)} ≈ log(n)^2.1 our analysis matches the lower bound 2^k (up to polynomial factors). Our bound on the degree greatly improves the previous result of Kolountzakis et al. [Combinatorica, 2009] who proved that |S| = O(k/ log k).