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

Abstract

We prove the uniform oscillation and jump inequalities for the polynomial ergodic averages modeled over multi-dimensional subsets of primes. This is a contribution to the Rosenblatt–Wierdl conjecture (Lond Math Soc Lect Notes 205:3–151, 1995, Problem 4.12, p. 80) with averages taken over primes. These inequalities provide endpoints for the r-variational estimates obtained by Trojan (Math Ann 374:1597–1656, 2019).