Fourier Entropy–Influence Conjecture

Prove that there exists a universal constant C such that for every Boolean function f: {−1,1}^n → {−1,1}, the spectral entropy H[ĥf^2] ≤ C · TInf[f], where H[ĥf^2] = ∑S ĥf(S)^2 log2(1/ĥf(S)^2) and TInf[f] is the total influence.

Background

This conjecture links the entropy of the Fourier spectrum of a Boolean function to its total influence, tying analytic regularity to combinatorial sensitivity. It has been verified for broad classes such as symmetric functions and read-once decision trees.

A weaker min-entropy variant implies KKL and is implied by KKL for monotone functions. Known lower bounds show any universal C must be at least 60/13 for some explicit functions.