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 Fouriercoefficient in any non-linear symmetric Boolean function. Specifically, we prove that forevery non-linear and symmetric $f:\B^{k}\to\B$ there exists a set $\emptyset\neqS\subset[k]$ such that $|S|=O(\Gamma(k)+\sqrt{k})$, and $\hat{f}(S)\neq0$, 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 learningalgorithm 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$n^{O(k^{0.525})}\cdot\poly(n,2^{k},\log(1/\delta))$ where $n$ is the number ofvariables, $k$ 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).\sloppy Our bound on the degree greatly improves the previous result of Kolountzakis etal. [Combinatorica, 2009] who proved that $|S|=O(k/\log k)$.