7 problems

Joint ergodicity conjecture. The polynomials $p_1,\ldots,p_\ell$ are jointly ergodic for $(X,\mathcal{X},\mu,U)$ if and only if: (1) the product action…

Let $\mathcal{H}$ denote the Hardy-field class used in the source, and let $[\ell]=\{1,\ldots,\ell\}$. For sequences $a_1,\ldots,a_\ell$ and a non-decreasing sequence $W(n)$, defin…

Let $(X,\mathcal{X},\mu,T_1,T_2)$ be a measure-preserving system. Joint ergodicity for logarithmic averages means convergence of the corresponding logarithmically weighted multiple…

Let $\ell\in\mathbb{N}$ and let $p_1,\ldots,p_\ell\in\mathbb{R}[x]$. For a family of real polynomials, $\mathbb{R}\backslash\mathbb{Q}_*$-independence means $V$-independence with…

Let $(X,\mathcal{B},\mu,T_{1},\dots,T_{d})$ be a system, and let … where $(c_i^1,c_i^2)\in\mathbb{R}^2\setminus\{(0,0)\}$, $h_i\in\mathcal{H}$, and $t_i\in\mathcal{T}$. Assume…

Let $(X,\mathcal X,\mu)$ be a probability space, let $T_1,\ldots,T_\ell$ be commuting measure-preserving transformations, and let $p_1,\ldots,p_\ell\in\mathbb Z[n]$. A sequence of…

Let $d,k,L\in\mathbb{N}$, let $p_{1},\dots,p_{k}\colon\mathbb{Z}^{L}\to\mathbb{Z}^{d}$ be polynomials, and let $(X,\mathcal{B},\mu,(T_{g})_{g\in\mathbb{Z}^{d}})$ be a…