15 problems

Let $p_1,\ldots,p_\ell\in\mathbb{Z}[t]$ be jointly intersective, meaning that for every $r\in\mathbb{N}$ there exists $n\in\mathbb{N}$ such that $r$ divides each of…

Let $(X,\mathcal{B},\mu)$ be a probability space, and let $T,S:X\to X$ be two not necessarily commuting measure-preserving transformations. Let $0<c<1/2$, and suppose that…

Multiple recurrence conjecture. This liminf is positive. Furthermore, if all the rational polynomials have degree $0$ and $R_j(m_0,n_0)=1$ for $j=1,\ldots,\ell$ for some…

Mean convergence conjecture. These averages converge in $L^2(\mu)$ as $N\to\infty$. Furthermore, if all the rational polynomials have degree $0$, then the conclusion holds for all…

Let $k\in\mathbb{N}$ and let $a_1,\ldots,a_k\in\mathcal{H}$ be functions of polynomial growth. Assume that every non-trivial linear combination $a$ of $a_1,\ldots,a_k$ satisfies ……

Let $\mathcal{H}$ be the Hardy field under consideration, let $a\in\mathcal{H}$ have polynomial growth, and assume … for every $c\in\mathbb{R}$ and $p(t)\in\mathbb{Z}[t]$. Let…

Multiple recurrence conjecture. The set $A$ contains a pattern of the form

Let $K$ be a degree $d$ number field with ring of integers $O_K\cong\mathbb{Z}^d$, and let $p(x)\in K[x]$ be an $O_K$-valued intersective polynomial. Finitary configuration conject…

Let $K$ be a number field with ring of integers $O_K$. Let $r,s\thisattacksim$ be distinct and nonzero. For an $O_K$-system $(X,\mathcal{B},\mu,T)$, an $O_K$-valued intersective po…

Parallelogram conjecture. The admissible triple $(r,s,t)$ has the large intersections property if and only if $(0,r,s,t)$ forms a parallelogram.

Quadruple-recurrence conjecture. If $k\ge4$, then $\{\varphi_1,\dots,\varphi_k\}$ does not have the large intersections property.

Non-ergodic failure conjecture. There exist a necessarily non-ergodic measure-preserving system $(X,\mathcal B,\mu,(T_g)_{g\in G})$, a set $A\in\mathcal B$ with $\mu(A)>0$, and…

Pair characterization conjecture. The pair $\{\varphi,\psi\}$ has the large intersections property if and only if it is admissible.

Let $p$ be a prime, let $k<p$, and let $c_0,\ldots,c_k$ be distinct elements of $\mathbb{F}_p$. A tuple has the Khintchine property when it satisfies the recurrence property define…

General Structural Conjecture. Depending only on these data, there are finite families of pairs $(\Lambda_{i,j},m_{i,j})$ for $i=1,2,\ldots,r$, with $m_{i,j}\geq 0$ and…