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

TL;DR

This paper extends classical ergodic theorems to multi-dimensional polynomial prime subsets by establishing $ ext{l}^p$-boundedness of variational seminorms for related discrete Radon operators, advancing understanding in ergodic theory and number theory.

Contribution

It introduces new $ ext{l}^p$-boundedness results for variational seminorms of discrete Radon-type operators on polynomial prime subsets, leading to extended ergodic theorems.

Findings

Proved $ ext{l}^p$-boundedness of variational seminorms for discrete operators

Extended Birkhoff's and Cotlar's ergodic theorems to polynomial prime subsets

Established convergence results for multi-dimensional polynomial prime configurations

Abstract

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 $r > 2$.