4 problems

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Bergelson–Leibman conjecture on polynomial ergodic averages for nilpotent actions
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…
Conjectureopenergodic-theory nilpotent-groups polynomial-averagesarXiv posted 2 days ago
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Bellow–Furstenberg pointwise convergence conjecture for multi-parameter polynomial averages
Bellow–Furstenberg conjecture. For $\mu$ -almost every $x\in X$ , these averages converge as $\min\{M_1,\ldots,M_k\}\to\infty$ .
Conjectureopenergodic-theory pointwise-convergence polynomial-averagesarXiv posted 2 days ago
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The degree-uniform improving conjecture for polynomial averages
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…
Conjectureopenharmonic-analysis lp-improving polynomial-averagesarXiv posted 2 days ago
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Unboundedness of polynomial ergodic averages' variation operator on $L^p$
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…
Conjectureopenergodic-theory polynomial-averages variation-normsarXiv posted 2 days ago

Bergelson–Leibman conjecture on polynomial ergodic averages for nilpotent actions