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

$f(x)=\sum _{S\subset [n]}\hat{f}(S)x_S,\text{ where }{x}_S=\prod _{i\in S}{x}_i.$

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

$I(f)=\sum _S|S|\hat{f}(S{)}^2,$

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

$H(f)=-\sum _S\hat{f}(S{)}^2\log \hat{f}(S{)}^2,$

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

See also

• Analysis of Boolean functions

References

• Unsolved Problems in Number Theory, Logic and Cryptography
• The Open Problems Project, discrete and computational geometry problems