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We prove the extensions of Birkhoff's and Cotlar's ergodic theorems tomulti-dimensional polynomial subsets of prime numbers $\mathbb{P}^k$. We deducethem from $\ell^p$-boundedness of $r$-variational seminorms for thecorresponding discrete operators of Radon type, where $p > 1$ and $r > 2$.

Variational estimates for discrete operators modeled on multi-dimensional polynomial subsets of primes