13 problems

Minimum Fourier entropy–influence conjecture. There exists a constant $c>0$ such that, for every $n$ and every Boolean function $f$,

Fourier-Min-Entropy-Influence conjecture. There exists a constant $c>0$ such that, for every Boolean function $f:\\{-1,1\\}^n\to\\{-1,1\\}$,

Fourier-Entropy-Influence conjecture. There exists a constant $c>0$ such that, for every Boolean function $f:\\{-1,1\\}^n\to\\{-1,1\\}$,

Friedgut–Kalai Fourier entropy-influence conjecture. There exists a universal constant $c$ such that, for every such $f$,

Quantum Fourier entropy-influence conjecture. There is a constant $c$ independent of $n$ such that

Let $f\colon [0,1]^n \to \{0,1\}$ be a measurable function, and let $I_S^b(f)$ denote the corresponding resilience quantity for a set $S\subseteq[n]$ and $b\in\{0,1\}$. Friedgut's…

Positive-influence conjecture. One has

Let $f\colon\left\{0,1\right\}^{n}\to\left\{0,1\right\}$ be a Boolean function, write $\mu=\mathbb{E}[f]$, and let $I(f)$ denote its total influence. A function $g$ is a Junta if i…

Let $f\colon\left\{0,1\right\}^{n}\to\left\{0,1\right\}$ be a monotone Boolean function with expectation $\mathbb{E}[f]=\mu$, and let $I[f]$ denote its total influence. A DNF of wi…

Fix $0<p<1$. Let $f:\{-1,1\}^n_p\to\{-1,1\}$ be a Boolean function, let $\operatorname{I}(f)$ denote its total influence, and let $\hat f(S)$ be its Fourier coefficients. Fourier c…

Let $\mu_p$ be the product measure on the discrete cube $\{0,1\}^n$, and let $f:\{0,1\}^n_p\to\{-1,1\}$ be a Boolean function. Write $\operatorname{Ent}_p(f)$ for its spectral entr…

Let $f:0,1n_{1/2}\to\{-1,1\}$ be a Boolean function on the discrete cube with uniform measure $\mu_{1/2}$. Write $\operatorname{Ent}_{1/2}(f)$ for the entropy of its Fourier-weight…

Keller–Pilpel's conjecture. The total influence of $f$ should satisfy