4 problems

Bergelson–Leibman conjecture. The limit of these averages exists in $L^2$-norm and almost everywhere. The conjecture concerns pointwise and norm convergence of polynomial multiple…

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

Let $d\geq 2$ be an integer, let $p(x)$ be any polynomial of degree $d$ mapping the integers to the integers, and let $N\geq 1$. Define … For the interval $I=[0,N^d]$, let $f$ be s…

Let $(X,\mathcal{B},\mu,T)$ be the measure-preserving system under consideration, let $P(n)$ be an integer-valued polynomial, and for $N\geq 1$ write … For a sequence $(a_N)$, let…