19 problems

Multiparameter polynomial pointwise ergodic conjecture. For every $f\in L^\infty(X)$, these averages converge for $\mu$-almost every $x\in X$ as $\min\{M_1,\ldots,M_k\}\to\infty$.…

Akcoglu–Jones–Rosenblatt convergence conjecture. The sequence $(1/M_N)S_Nf$ is convergent if and only if

For $R>0$, let $S_R^{\alpha}$ denote the Bochner–Riesz means on $\mathbb{R}^n$. Almost-everywhere convergence conjecture. When $p\geq2$, almost-everywhere convergence … for…

Let $\mathbb{T}$ be the unit circle, let $dm$ denote normalized Lebesgue measure on $\mathbb{T}$, and let $L^2(\mathbb{T})$ be the space of square-integrable functions on…

For each prime $l$, let $v_l(t)$ denote the local factor defined from the trace distribution in $\operatorname{GSp}_{2g}(\mathbb{Z}_l)$, with $v_p(t)=1$, and let…

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

Multiplicative-weighted polynomial multiple ergodic averages conjecture. For almost all $x\in X$, the limit of these averages exists as $N\to\infty$. This is presented as the polyn…

Möbius-weighted polynomial multiple ergodic averages conjecture. For almost all $x\in X$, these averages converge to $0$ as $N\to\infty$. This is motivated by the Möbius randomness…

Let $(X,\mathcal{X},\mu,T)$ be a measure-preserving system, let $a_1,\ldots,a_m\in\mathbb{Z}$ and $f_1,\ldots,f_m\in L^\infty(X,\mu)$, and let $(\Psi(n))_{n\in\mathbb{Z}}$ be a nil…

Polynomial pointwise convergence conjecture. These averages converge almost surely. This is a natural extension of Bourgain's double recurrence theorem; recent progress is known, b…

Multi-parameter linear polynomial convergence conjecture. For every $f\in L^{\infty}(X)$, these averages converge for $\mu$-almost every $x\in X$ as…

Multi-parameter Bellow–Furstenberg conjecture. For every $f\in L^{\infty}(X)$, the limit

Bellow–Furstenberg conjecture. For $\mu$-almost every $x\in X$, these averages converge as $\min\{M_1,\ldots,M_k\}\to\infty$.

The $1/4$-regularity conjecture. The minimal value of $s$ for which this convergence holds for every $f\in H^s(\mathbb{R}^n)$ should be $s=1/4$ in all dimensions. This was a long-s…

Let $n$ be the spatial dimension, let $s$ denote the Sobolev regularity, and let $alpha=n$ be the full-dimensional Hausdorff-measure case for the pointwise convergence problem for…

Referee's conjecture. The arguments and proof methods should also work for Dunford–Schwartz operators, not just for Koopman operators.

Let $\mathbb{T}$ be the one-dimensional torus, let $L\log L(\mathbb{T})$ denote the corresponding Orlicz space, and for $f\in L\log L(\mathbb{T})$ let $S_n(f)$ be the $n$th partial…

Let $\{n_j:j\ge 1\}$ be a lacunary sequence of integers, meaning that … For a function $f$ on $\mathbb T$, let $S_{n_j}f$ denote its Walsh-Fourier partial sums, and write…

Positive-measure divergence conjecture. There exists a sequence of probability measures $\{\mu_n\}$ such that