Double Recurrence and Almost Sure Convergence: Primes and Weighted Theory

Let $(X,\mu )$ be a probability space equipped with an invertible, measure-preserving transformation $T\text{\hspace{0.17em}⁣}:X\to X$. We exhibit a wide class of weights $w$ so that whenever $f,g\in L^{\mathrm{\infty }}(X)$, the bilinear ergodic averages $$\frac{1}{N}\sum _{n\le N}w(n)T^{an}f\cdot T^{bn}g,a,b\in \mathbb{Z}$$ 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 $\{\lfloor k^c\rfloor :k\in \mathbb{N}\}$, $1\le c<7\mathrm{/}6$. 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.