Pointwise convergence of bilinear polynomial averages over the primes

Abstract

We show that on a $\sigma$-finite measure preserving system $X=(X,\nu ,T)$, the non-conventional ergodic averages $${\mathbb{E}}_{n\in [N]}\Lambda (n)f(T^{n}x)g(T^{P(n)}x)$$ converge pointwise almost everywhere for $f\in L^{p_1}(X)$, $g\in L^{p_2}(X)$, and $1/p_1+1/p_2\le 1$, where $P$ is a polynomial with integer coefficients of degree at least $2$. This had previously been established with the von Mangoldt weight $\Lambda$ replaced by the constant weight $1$ by the first and third authors with Mirek, and by the Möbius weight $\mu$ by the fourth author. The proof is based on combining tools from both of these papers, together with several Gowers norm and polynomial averaging operator estimates on approximants to the von Mangoldt function of ''Cramér'' and ''Heath-Brown'' type.

arXiv comment38 pages; referee comments incorporated

References in corpus (6)

• Variational estimates for averages and truncated singular integrals along the prime numbers
• Bounds for sets with no polynomial progressions
• Higher uniformity of arithmetic functions in short intervals I. All intervals
• Improving estimates for discrete polynomial averages
• Quantitative bounds for Gowers uniformity of the Möbius and von Mangoldt functions
• Polynomial progressions in topological fields