Fourier-Entropy-Influence (FEI) Conjecture on the uniform hypercube

Establish the Fourier-Entropy-Influence inequality on the uniform cube by proving that there exists a universal constant C > 0 such that for every n and every Boolean function f: ( {0,1}^n, μ_{1/2}^n ) → {−1,1}, the spectral entropy satisfies Ent_{1/2}(f) ≤ C · I^{(1/2)}[f], where I^{(1/2)}[f] denotes the total influence under the uniform measure.

Background

The Fourier-Entropy-Influence (FEI) conjecture seeks a universal upper bound on the spectral (Fourier) entropy of Boolean functions on the uniform hypercube in terms of their total influence. It was proposed by Friedgut and Kalai (1996) and is regarded as a central open problem in the analysis of Boolean functions, connecting spectral concentration to combinatorial sensitivity.

The paper discusses spectral entropy and influences on biased cubes and references the FEI conjecture as a motivating upper-bound direction, contrasting it with the new lower bounds proved herein. The conjecture has driven substantial research, with partial progress known for special classes of functions but no general resolution.

References

In the direction of upper bounds on Fourier entropy, it is natural to recall the original Fourier-Entropy-Influence (FEI) Conjecture of Friedgut and Kalai in 1996, one of the longstanding and influential open problems in the field: There exists a universal constant $C>0$ such that for every $n$ and every Boolean function $f:({0,1}n,\mu_{1/2}n)\to{\pm 1}$, ${\rm Ent}_{1/2}(f) \le C\cdot {\rm I}{(1/2)}[f]."

— A Lower Bound for the Fourier Entropy of Boolean Functions on the Biased Hypercube (2511.07739 - Chang, 11 Nov 2025) in Section 1, Introduction (Conjecture [Friedgut–Kalai])