Four equivalent forms of the Fourier entropy-influence conjecture

Let f be a Boolean function with spectral distribution μ(S)= f(S)², and let γ(T)=∑U⊇ Tμ(U) be the generalized influence of a set T. The Fourier entropy splits as $H(f)=A+B$ with A=E_μlog₂(1/γ(S)) and B=E_μlog₂(γ(S)/μ(S))≥ 0. Since derivatives of Boolean functions take values in a lattice, A≤ H(f)/2+I(f)-Var(f), and so the Fourier entropy-influence conjecture is equivalent to B≤ C\,I(f). We give three further equivalent forms: a bound on the cross-entropy of the spectral distribution against the noisy generalized influences \|T√ t\,∂T f\|₂² at a fixed $t<1$; a bound on the integral over t∈[12,1] of the expected number of elements a noise-tilted posterior adds to the spectral sample; and a bound of $O(1)$ bits on the log-loss of predicting a uniformly chosen element of the size-biased spectral sample from the others, which also has a variational form over all predictors. Form (c) gives the unconditional bound H(f)≤ 4I(f)-2Var(f)+2M(f), where $M(f)$ is the expected number of such added elements at $t=1$.