Physics:The Entropy Influence Conjecture

This article describes The Entropy Influence Conjecture.

The Conjecture

For a function $f:\{-1,1{\}}^n\to \{-1,1\}$ the Entropy-Influence relates the following two quantities, both of which may be expressed in terms of the Fourier Expansion of the functions $f=\sum _{S\subset [n]}\hat{f}(S)x_S$, where $x_S=\prod _{i\in S}{x}_i$. The first expression is the total influence of the function defined by $I(f)=\sum _S|S|{\hat{f}}^2(S)$. The second terms is the Entropy (of the spectrum) of the function defined by $H(f)=-\sum _S{\hat{f}}^2(S)\log {\hat{f}}^2(S)$ (where $x\log x=0$ when $x=0$).

The conjecture states that there exists an absolute constant C such that for all Boolean functions $f:\{-1,1{\}}^n\to \{-1,1\}$ it holds that $H(f)\le CI(f)$.

See also

• Hilbert's 23 problems

References

• Cameron–Erdős conjecture (Ben J. Green, 2003, conjectured by Paul)

References

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