Some open problems on multiple ergodic averages

TL;DR

This paper surveys recent advances and open problems in multiple recurrence and convergence of $\

Contribution

It compiles recent developments and highlights unresolved questions in multiple ergodic averages for $\

Findings

Identifies key open problems in multiple recurrence.

Summarizes recent progress in convergence phenomena.

Connects ergodic theory with combinatorics and number theory.

Abstract

We survey some recent developments and give a list of open problems regarding multiple recurrence and convergence phenomena of $\mathbb{Z}^d$ actions in ergodic theory and related applications in combinatorics and number theory.