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.

References

Weaker version: the “Min-Entropy--Influence Conjecture”, which states that there exists S such that ĥf(S)2 ≥ 2{-C * TInf[f]}.

— Open Problems in Analysis of Boolean Functions  (1204.6447 - O'Donnell, 2012) in Main matter, problem “Fourier Entropy–Influence Conjecture,” remarks, fourth bullet

The conjecture asserts the existence of a universal constant $C$ such that $H(f)\leq C \cdot I(f)$ for every Boolean function $f$.

— On the Fourier Entropy-Influence Conjecture for Boolean Plateaued Functions  (2609.21563 - Potapov, 18 Sep 2026) in Section 1, Introduction