Oscillation and jump inequalities for the polynomial ergodic averages along multi-dimensional subsets of primes

TL;DR

This paper establishes uniform oscillation and jump inequalities for polynomial ergodic averages over multi-dimensional prime subsets, extending variational estimates and providing new endpoint results in ergodic theory.

Contribution

It introduces novel uniform inequalities for polynomial ergodic averages over primes, advancing the understanding of variational estimates in ergodic theory.

Findings

Proves uniform oscillation inequalities for polynomial ergodic averages.

Establishes jump inequalities for averages over multi-dimensional prime subsets.

Provides endpoint results for r-variational estimates.

Abstract

We prove the uniform oscillation and jump inequalities for the polynomial ergodic averages modeled over multi-dimensional subset of primes. These inequalities provide endpoints for the $r$-variational estimates obtained by Trojan arXiv:1803.05406.