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 nonlinear symmetric Boolean function. Specifically, we prove that for every non-linear and symmetric f: {0,1}^k →{0,1} there exists a set θ ≠ S ⊂ [k] such that |S| = O(Λ(k) + √k), and f̂(S) ≠ 0, where Λ(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/logk).