Double Recurrence and Almost Sure Convergence: Primes and Weighted Theory

Let $(X,\mu)$ be a probability space equipped with an invertible, measure-preserving transformation $T\colon X \to X$. We exhibit a wide class of weights $w$ so that whenever $f,g \in L^{\infty}(X)$, the bilinear ergodic averages $$\frac{1}{N} \sum_{n \leq N} w(n)\, T^{an}f \cdot T^{bn}g, \qquad a,b \in \mathbb{Z}$$ converge $\mu$-almost surely. This class encompasses the von Mangoldt function, resolving Problem 12 from Frantzikinakis' survey on open problems in ergodic theory, the divisor function, the sum-of-two-squares representation function, etc., as well as their restrictions to lower-density Piatetski-Shapiro sequences of the form $\{\lfloor k^{c}\rfloor : k \in \mathbb{N}\}$, $1 \leq c < 7/6$. Our methods combine combinatorial number theory and higher-order Fourier analysis with classical Fourier-analytic/martingale-based methods; the role of $U^{3}$ analysis is particularly significant.