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 $f:\{0,1\}^{k} \to \{0,1\}$ there exists a set $\emptyset\neq S\subset[k]$ such that $|S|=O(\Gamma(k)+\sqrt{k})$, and $\hat{f}(S) \neq 0$, where $\Gamma(m) \leq m^{0.525}$ is the largest gap between consecutive prime numbers in $\{1,\ldots,m\}$.
As an application we obtain a new analysis of the PAC learning algorithm for symmetric juntas, under the uniform distribution, of Mossel et al. Namely, we show that the running time of their algorithm is at most $n^{O(k^{0.525})} \cdot \mathrm{poly}(n,2^{k},\log(1/\delta))$ where $n$ is the number of variables, $k$ is the size of the junta (i.e. number of relevant variables) and $\delta$ is the error probability. In particular, for $k\geq\log(n)^{1/(1-0.525)}\approx \log(n)^{2.1}$ 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. who proved that $|S|=O(k/\log k)$.