Pointwise convergence of polynomial multiple ergodic averages along the primes

We establish pointwise almost everywhere convergence for the polynomial multiple ergodic averages $$\frac{1}{N}\sum _{n=1}^N\text{La}(n)f_1(T^{P_1(n)}x)\cdots f_k(T^{P_k(n)}x)$$ as $N\to \mathrm{\infty }$, where $\text{La}$ is the von Mangoldt function, $T\text{\hspace{0.17em}⁣}:X\to X$ is an invertible measure-preserving transformation of a probability space $(X,\nu )$, $P_1,\dots ,P_k$ are polynomials with integer coefficients and distinct degrees, and $f_1,\dots ,f_k\in L^{\mathrm{\infty }}(X)$. This pointwise almost everywhere convergence result can be seen as a refinement of the norm convergence result obtained in Wooley–Ziegler (Amer. J. Math, 2012) in the case of polynomials with distinct degrees. Building on the foundational work of Krause–Mirek–Tao (Ann. of Math., 2022), Kosz–Mirek–Peluse–Wright (arXiv: 2411.09478, 2024), and Krause–Mousavi–Tao–Teräväinen (arXiv: 2409.10510, 2024), we develop a multilinear circle method for von Mangoldt-weighted (equivalently, prime-weighted) averages. This method combines harmonic analysis techniques across multiple groups with the newest inverse theorem from additive combinatorics. In particular, the principal innovations of this framework include: (i) an inverse theorem and a Weyl-type inequality for multilinear Cramér-weighted averages; (ii) a multilinear Rademacher-Menshov inequality; and (iii) an arithmetic multilinear estimate.