In this thesis 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 non-empty subset S of {1,...,n} such that |S| = O(Γ(k) √k), and the S-fourier coefficient of f is non-zero, 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. [MOS04]. Namely, we show that the running time of their algorithm is at most n^{O(Γ(k) √k)} 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. [KLMMV09] who proved that |S| = O(k / log(k)).