11 problems

Let $P=(p_1,\b7dots,p_r)$ be a system of rationally independent polynomials, and let $Z_k(X)$ denote the factor appearing in the definition of an $\b7IP\b7rat$-characteristic facto…

Ergodic polynomial odd-recurrence conjecture. The set $R$ has nonempty intersection with every infinite arithmetic progression of step size $k$.

Let $P=\{p_1,\ldots,p_k\}$ and $Q=\{q_1,\ldots,q_l\}$ be sets of polynomials. Let $\textup{\textsf{{WP}}}(P)$ and $\textup{\textsf{{WP}}}(Q)$ denote their associated Weyl-polynomia…

Generalized orthogonality conjecture.

Let $P$ be a set of integral polynomials, and let $W(P)$ denote its Weyl complexity. Bergelson–Leibman–Lesigne conjecture. … A better understanding of the stabilization behavior of…

Let $p$ and $q$ be integral polynomials vanishing at $0$, with $\deg p\ge 2$ and $\deg q\ge 2$. Let $(X,T)$ and $(X,S)$ be systems on the same compact space $X$. Polynomial simulta…

Let $(X,T^k)$ be minimal for some $k\geq 2$, let $d\in\mathbb N$, and let $P_m(n)$, $1\leq m\leq d$, be non-constant integral polynomials satisfying $P_m(0)=0$. Simultaneous polyno…

Let $(X,T)$ be a totally minimal system, and let $P(n)$ be a non-constant integral polynomial. Polynomial recurrence conjecture. There is a dense $G_\delta$ subset $\Omega$ of $X$…

Ring-condition conjecture. If the ring generated by $p_1,\ldots,p_r$ contains no nonzero constants, then $P$ has the multiple recurrence property in totally ergodic systems. The co…

Let $p_1,\ldots,p_r$ be integral polynomials in one variable. The natural conjecture that these polynomials are jointly intersective if every linear combination of them is intersec…

Let $P$ be a set of primes, and write $\overline{d}_{\mathcal P}(P)$ for its relative upper density within the primes. Let $\psi_1(x),\ldots,\psi_m(x)$ be polynomials with rational…