A New Bound for the Fourier-Entropy-Influence Conjecture

Abstract

In this paper, we prove that the Fourier entropy of an n-dimensional boolean function f can be upper-bounded by \(O(I(f)+ \sum \limits _{k\in [n]}I_k(f)\log \frac{1}{I_k(f)})\), where I(f) is its total influence and \(I_k(f)\) is the influence of the k-th coordinate. There is no strict quantitative relationship between our bound with the known bounds for the Fourier-Min-Entropy-Influence conjecture \(O(I(f)\log I(f))\) and \(O(I(f)^2)\). The proof is elementary and uses iterative bounds on moments of Fourier coefficients over different levels to estimate the Fourier entropy as its derivative.