OSCILLATION AND JUMP INEQUALITIES FOR THE POLYNOMIAL ERGODIC AVERAGES ALONG MULTI-DIMENSIONAL SUBSETS OF PRIMES

The paper proves uniform L^p jump and oscillation inequalities for polynomial ergodic averages and Cotlar-type transforms along multi-dimensional subsets of primes by (i) reducing to the integer shift model via Calderón transference, and (ii) establishing discrete ℓ^p jump/oscillation bounds via the circle method, Ionescu–Wainger theory, and sampling/rademacher–Menshov techniques. The candidate solution outlines exactly this blueprint: transference, uniform discrete bounds (with logarithmic prime weights) independent of polynomial coefficients, then transference back. The only gap in the model sketch is that it treats the discrete estimates as a black box—precisely the main technical contribution of the paper. Within the scope of reconciliation, both are correct and follow substantially the same proof strategy.