12 problems

Rosenblatt–Wierdl conjecture. does not converge almost everywhere. This conjecture concerns divergence of sparse ergodic averages at the $L^1$ endpoint; the paper studies quantitat…

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…

The $k$-linear pointwise convergence conjecture. Every measure-preserving system satisfies $k$-linear pointwise convergence.

Let $(X,\mathcal{B},\mathbb{P},T)$ be a measure-preserving system, and let $\mu_N^T f=\sum_{n\in\mathbb{Z}}\mu_N(n)f\circ T^n$ be a dissipative sequence of operators. Assume that ……

Let $theta otinboldsymbol{Q}$ and let $4mathsf{x}={xn}n$ with $xn=ntheta$. For a continuous function $f$, write … We say that these averages fluctuate when they are infinitely ofte…

Double Birkhoff averages conjecture. Every $\mathbb{R}^{m}$-system is good for double Birkhoff averages: for every $f_1,f_2\in L^{\infty}(\mu)$ and every…

Let $\bm \nu$ be an aperiodic bounded multiplicative function and let $l\geq 2$ be a positive integer. Let $T_1,\ldots,T_l$ be commuting measure-preserving transformations acting o…

Divergence conjecture. There exists a function $f\geq 0$ such that

Let $p_1,\ldots,p_\ell\in\mathbb{Z}[t]$ be the polynomial iterates, and let the conclusion of Theorem denote the asserted characteristic-factor property for the associated multiple…

Let $(s_n)$ be a sequence of integers. An $\ell$-step nilsystem is a measure-preserving system arising from a translation on a compact homogeneous space $G/\Gamma$, where $G$ is an…

Let $(s_n)$ be a sequence of integers, and let $k,\ell\in\mathbb{N}$. Power-transfer conjecture. If $(s_n)$ is good for mean $(k\ell)$-convergence, then $(s_n^k)$ is good for…

Let $(s_n)$ be a sequence of integers, and let $\boldsymbol{\text{good for }\ell\text{-convergence}}$ mean that the sequence has the corresponding multiple ergodic convergence prop…