Entropy influence conjecture

In mathematics, the entropy influence conjecture is a statement about Boolean functions originally conjectured by Ehud Friedgut and Gil Kalai in 1996.

Statement
For a function $$ f: \{-1,1\}^n \to \{-1,1\},$$ note its Fourier expansion


 * $$ f(x) = \sum_{S \subset [n]} \widehat{f}(S) x_S, \text{ where } x_S = \prod_{i \in S} x_i. $$

The entropy–influence conjecture states that there exists an absolute constant C such that $$H(f) \leq C I(f),$$ where the total influence $$I$$ is defined by


 * $$ I(f) = \sum_S |S| \widehat{f}(S)^2, $$

and the entropy $$H$$ (of the spectrum) is defined by


 * $$ H(f) = - \sum_S \widehat{f}(S)^2 \log \widehat{f}(S)^2 ,$$

(where x log x is taken to be 0 when x = 0).