Multiple ergodic averages — Problem 12

Definitions and mathematical statement

Let $T$ be an invertible measure-preserving transformation of a probability space $(X,\mathcal B,\mu)$, and let $f,g\in L^\infty(\mu)$. Define the von Mangoldt function by $\Lambda(p^k)=\log p$ for primes $p$ and integers $k\ge1$, and $\Lambda(n)=0$ otherwise. Also let $\varphi:\mathbb N\to\mathbb C$ satisfy $|\varphi(n)|\le1$ and $\varphi(mn)=\varphi(m)\varphi(n)$ when $\gcd(m,n)=1$. Assume it has convergent means on every arithmetic progression: \[ \lim_{N\to\infty}\frac1N\sum_{n=1}^N \varphi(an+b)\quad\text{exists for all }a,b\ge1. \] The two requested conclusions are almost-everywhere convergence of \[ \frac1N\sum_{n=1}^N\Lambda(n)f(T^n x)g(T^{2n}x), \qquad \frac1N\sum_{n=1}^N\varphi(n)f(T^n x)g(T^{2n}x). \] Each assertion is universal over the indicated systems and functions; the second is also universal over the weights just defined. Convergence is along all positive integers $N$, to a finite complex limit which need not be specified in advance.

The progression assumption is stronger than convergence of the single unweighted mean of $\varphi(n)$. Multiplicativity here is only required for coprime arguments, rather than complete multiplicativity. No ergodicity is assumed. The author's update records progress for particular weights; this formulation retains both the von Mangoldt target and the full class of multiplicative weights in the original problem.

Short English statement

Prove pointwise convergence of double linear ergodic averages weighted by von Mangoldt or by bounded multiplicative functions with progression means.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4800012 (AMR-047-0012), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Frantzikinakis - Some open problems on multiple ergodic averages (2016), Problem 12. Original source indexed by AMR-047-0012. https://arxiv.org/abs/1103.3808.