POINTWISE CONVERGENCE OF BILINEAR POLYNOMIAL AVERAGES OVER THE PRIMES

The uploaded paper proves exactly the two statements the model claims. Theorem 1.3 establishes: (a) pointwise a.e. convergence for AN,Λ;X(f,g) on σ-finite systems for 1/p1+1/p2≤1, and (b) the lacunary r-variation estimate (1.9) for any finite λ-lacunary set D and r>2 with 1/p=1/p1+1/p2; the range r>2 is stated as optimal. This matches the model’s (a) and (b) verbatim. The paper’s methods combine the 2022 unweighted bilinear theory with Λ-approximants (Cramér/Heath–Brown), Gowers-uniformity bounds, transference, and p-adic estimates, just as the model summarized. Minor nit: the model called the main statement “Theorem 1.1,” while in the PDF it is Theorem 1.3. Otherwise, no missing hypotheses or contradictions were found. See Theorem 1.3 and (1.9) for the precise statement, and the definition/setup for σ-finite systems and the operators AN,w;X(f,g) in the introduction, together with the high-level proof outline and remarks on optimality and slight extensions beyond duality in Section 6.