Upper bounds on Fourier entropy

Given a function \(f : {\{0,1\}}^n\rightarrow \mathbb {R}\), its Fourier Entropy is defined to be \(-\sum _S {\widehat{f}}^2(S) \log {\widehat{f}}^2(S)\), where \(\hat{f}\) denotes the Fourier transform of f. This quantity arises in a number of applications, especially in the study of Boolean functions. An outstanding open question is a conjecture of Friedgut and Kalai (1996), called the Fourier Entropy Influence (FEI) Conjecture, asserting that the Fourier Entropy of any Boolean function f is bounded above, up to a constant factor, by the total influence (= average sensitivity) of f.