22 problems

Let $T,S\in\operatorname{SL}(d,\mathbb{Z})$, and let $\mathcal{D}$ be a Borel subset of $\mathbb{T}^d$ of positive measure. Boshernitzan's conjecture. … This is a measurable analog…

Let $f\colon\mathbb{N}\to\mathbb{R}$ be a tempered function, let $(X,\mathscr{B},\mu,T)$ be an invertible probability measure-preserving system, let $k\in\mathbb{N}$, let…

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.

Admissibility conjecture. Imposing admissibility on the homomorphisms is necessary to recover large intersections for the pair $\{\varphi,\psi\}$.

Multiple recurrence conjecture. If $z\in P$ is ML-random, then there exists $n$ such that

Let $G$) be a countable amenable group with a left Følner sequence $hi$. Let $T_1,ots,T_k$ be commuting, measure-preserving actions of $G$ on a probability space $(X,athscr{B},u)$.…

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…

Let $G$ be an ultra quasirandom group and let $A$ be the exceptional set appearing in conclusion (iii) of the proposition under discussion. Uniform expansion conjecture. The set…

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…

Let $T_1,\ldots,T_\ell$ be commuting, invertible measure-preserving transformations. Consider the lower-bound result established in the paper for the case in which all transformati…