The List-Decoding Size of Fourier-Sparse Boolean Functions

A function defined on the Boolean hypercube is k-Fourier-sparse if it has atmost knonzero Fourier coefficients. For a function f : F-2(n) -> R and parameters k and d, we prove a strong upper bound on the number of k-Fourier-sparse Boolean functions that disagree with f on at most d inputs. Our bound implies that the number of uniform and independent random samples needed for learning the class of k-Fourier-sparse Boolean functions on n variables exactly is at most O(n.k log k). As an application, we prove an upper bound on the query complexity of testing Booleanity of Fourier-sparse functions. Our bound is tight up to a logarithmic factor and quadratically improves on a result due to Gur and Tamuz [2013].