Friedgut–Kahn–Kalai–Keller spectral correlation conjecture

We work on the discrete cube $\{0,1\}^n$ with the uniform product measure. For functions $f,g$ on the cube, write

$$\operatorname{Cov}(f,g)=\mathbb{E}[fg]-\mathbb{E}[f]\mathbb{E}[g].$$

A function is increasing if $x\leq y$ coordinatewise implies $f(x)\leq f(y)$. For a Boolean function $f$, let $\hat f(S)$ denote its Fourier coefficient on $S\subseteq[n]$.

Friedgut–Kahn–Kalai–Keller spectral correlation conjecture. For any increasing Boolean functions $f,g:\{0,1\}^n\to\{0,1\}$,

$$\operatorname{Cov}(f,g)\geq 4\sum_{S\neq\emptyset}|S|\hat f(S)^2\hat g(S)^2.$$

This conjecture is a spectral strengthening of the Harris–Kleitman inequality and would sharpen Talagrand’s correlation theory. The paper proves the same inequality with constant $2$, while the conjectured constant $4$ is not resolved here; related structured cases were previously verified by Chang and Chen.