Fourier-Min-Entropy-Influence Conjecture (uniform hypercube)

Determine whether the min-entropy of the squared Fourier coefficient distribution of a Boolean function on the uniform hypercube can be bounded by a universal constant times its total influence; specifically, decide if min_{S⊂[n]} log_2(\hat f(S)^2) ≤ C · I^{(1/2)}[f] holds for all Boolean f: ( {0,1}^n, μ_{1/2}^n ) → {−1,1}.

Background

This conjecture is a weaker variant of the FEI conjecture, replacing Fourier entropy with min-entropy (a measure of the largest Fourier mass). It asks for a constant-factor relationship between min-entropy and total influence for Boolean functions under the uniform measure.

The paper mentions this conjecture to contextualize upper-bound efforts on spectral quantities and contrasts them with the lower-bound results proved in the biased setting.