10 problems

The multislice consists of vectors in $\{0,\ldots,m-1\}^n$ with prescribed numbers of coordinates of each value. More precisely, for a partition $n=\lambda_0+\cdots+\lambda_{m-1}$,…

Let $K \subseteq \mathbb{R}^n$ be a convex symmetric set with total convex influence $\operatorname{Inf}[K] \leq I$. For an accuracy parameter $\varepsilon>0$, consider orthonormal…

Verbin–Servedio–Tan conjecture. If $f$ is monotone and $\operatorname{DT}(f)=d$, then $f$ must be $\varepsilon$-close to a $\operatorname{poly}_{\varepsilon}(d)$-junta. This would…

Friedgut's narrow DNF conjecture. Every monotone function that has a coarse threshold is approximable by a narrow DNF.

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…

Independent junta conjecture. There should exist a set $T$ as above that is not only sparse in the product graph of dimension $j$, but actually independent; equivalently, any indep…

Let $f=(f_1,\ldots,f_m):\mathbb{Z}_k^n\to\mathbb{Z}_l^m$ be $\alpha$-Lipschitz with respect to the $L^1$ norm. Coordinatewise junta conjecture. For every $\delta,\epsilon>0$, at le…

Let $f:\mathbb{Z}_k^n\to\mathbb{Z}_l$ satisfy … Here $|\cdot|'$ is the source's normalized difference notation. Junta-size conjecture. The function $f$ is $\epsilon$-close to an…

Let $A\subset [k]^n$, and let $f=1_A$. For a fibre restriction $f_j^x:[k]\to\{0,1\}$, define $h(F)=0$ if $F$ is constant and $h(F)=1$ otherwise. The associated junta assertion is t…

Dinur–Friedgut conjecture. There exists a constant $c>0$ such that there is a set $J\subset\{1,2,\ldots,n\}$ with $|J|\leq\exp(cB/\epsilon)$ and a function…