On the Minimal Fourier Degree of Symmetric Boolean Functions

Abstract

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 :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(k 0.525 ) ·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. [KLM + 09] who proved that |S| = O(k/ log k).

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