Pointwise convergence of bilinear polynomial averages over the primes

TL;DR

The paper proves pointwise almost everywhere convergence of bilinear polynomial ergodic averages weighted by the von Mangoldt function, $\mathrm{A}_{N,\Lambda;X}(f,g)$, on $\sigma$-finite measure-preserving systems for exponents $p_1,p_2$ with $\frac{1}{p_1}+\frac{1}{p_2}\leq1$ and polynomials $P$ of degree at least 2. The authors synthesize the unweighted (KMT0) and Möbius-weighted (joni) frameworks, introducing scale-dependent approximants to $\Lambda$ (the Cramér and Heath–Brown models) to handle zeros of Dirichlet $L$-functions, Siegel zeros, and major/minor arc analysis, including p-adic and adelic components. They establish a variational ergodic theorem for $\Lambda$-weighted bilinear averages with sharp $r>2$ variation, and derive a corollary giving pointwise convergence for primes via the standard relation between $\log n$ and $\log N$. The work introduces a flexible approximation pipeline, along with novel bilinear major-arc machinery on adelic rings and weighted inverse theorems, broadening the scope of prime-weighted ergodic results and suggesting applicability to other arithmetically structured weights. Overall, this advances the understanding of how primes modulate nonlinear ergodic averages and provides robust tools for future investigations of Prime-based ergodic phenomena.

Abstract

We show that on a $σ$-finite measure preserving system $X = (X,ν, T)$, the non-conventional ergodic averages $$ \mathbb{E}_{n \in [N]} Λ(n) f(T^n x) g(T^{P(n)} x)$$ converge pointwise almost everywhere for $f \in L^{p_1}(X)$, $g \in L^{p_2}(X)$, and $1/p_1 + 1/p_2 \leq 1$, where $P$ is a polynomial with integer coefficients of degree at least $2$. This had previously been established with the von Mangoldt weight $Λ$ replaced by the constant weight $1$ by the first and third authors with Mirek, and by the Möbius weight $μ$ 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.

Theorems & Definitions (38)

• Theorem 1.1: Unweighted ergodic theorem
• Theorem 1.2: Möbius-weighted ergodic theorem
• Theorem 1.3: Main theorem
• Corollary 1.4
• Remark
• Remark
• Lemma 3.1: From $\Lambda$ to $\Lambda_N$
• proof
• Theorem 3.2: Single scale minor arc estimate
• Theorem 3.3: Variational paraproduct estimates