Prime-weighted polynomial multiple ergodic averages converge pointwise almost everywhere for arbitrary k and distinct-degree integer polynomials, with r-variational estimates for every r > 2.
Bergelson, Ergodic Ramsey Theory – an update, Ergodic Theory of Zd-actions (edited by M
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Pointwise convergence of polynomial multiple ergodic averages along the primes
Prime-weighted polynomial multiple ergodic averages converge pointwise almost everywhere for arbitrary k and distinct-degree integer polynomials, with r-variational estimates for every r > 2.