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

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