On the Fourier spectrum of symmetric Boolean functions

We study the following question: What is the smallest t such that every symmetric boolean function on κ variables (which is not a constant or a parity function), has a non-zero Fourier coefficient of order at least 1 and at most t? We exclude the constant functions for which there is no such t and the parity functions for which t has to be κ. Let τ (κ) be the smallest such t. Our main result is that for large κ, τ (κ)≤4κ/logκ. The motivation for our work is to understand the complexity of learning symmetric juntas. A κ-junta is a boolean function of n variables that depends only on an unknown subset of κ variables. A symmetric κ-junta is a junta that is symmetric in the variables it depends on. Our result implies an algorithm to learn the class of symmetric κ-juntas, in the uniform PAC learning model, in time no(κ). This improves on a result of Mossel, O'Donnell and Servedio in [16], who show that symmetric κ-juntas can be learned in time n2κ/3. © 2009 János Bolyai Mathematical Society and Springer Verlag.