POINTWISE CONVERGENCE OF POLYNOMIAL MULTIPLE ERGODIC AVERAGES ALONG THE PRIMES

The uploaded paper’s Theorem 1.1 states exactly the two claims: (i) pointwise a.e. convergence of the von Mangoldt–weighted multilinear polynomial ergodic averages with distinct degrees, and (ii) a lacunary r-variational L^q bound for every r>2 and 0<q<∞, along a λ-lacunary set D, matching the candidate’s assertions word-for-word . The paper reduces to the integer shift system via Calderón transference (Theorems 1.2–1.3) and proves the variational bound by a circle-method major/minor arc decomposition, minor-arc control through a generalized von Neumann theorem for prime-type (Cramér/Heath–Brown) approximants, and major-arc analysis using an Ionescu–Wainger-type discrete-to-continuous transference together with a multilinear Rademacher–Menshov inequality—precisely the strategy summarized by the model . The paper explicitly notes that (ii) implies (i), consistent with the model’s final step from lacunary variation to full pointwise convergence .