On a multi-parameter variant of the Bellow-Furstenberg problem

TL;DR

This paper establishes convergence results for multi-parameter polynomial ergodic averages in $L^p$ spaces, using novel multi-parameter oscillation inequalities and exponential sum estimates, addressing a multi-parameter Bellow-Furstenberg problem.

Contribution

It provides the first systematic treatment of multi-parameter oscillation semi-norms for pointwise convergence with arithmetic features, introducing new methods from the multi-parameter circle method.

Findings

Proves convergence in norm and almost everywhere for certain multi-parameter polynomial averages.

Develops multi-parameter oscillation inequalities to handle pointwise convergence.

Introduces new estimates for multi-parameter exponential sums and geometric analysis techniques.

Abstract

We prove convergence in norm and pointwise almost everywhere on $L^p$, $p\in (1,\infty)$, for certain multi-parameter polynomial ergodic averages by establishing the corresponding multi-parameter maximal and oscillation inequalities. Our result, in particular, gives an affirmative answer to a multi-parameter variant of the Bellow-Furstenberg problem. This paper is also the first systematic treatment of multi-parameter oscillation semi-norms which allows an efficient handling of multi-parameter pointwise convergence problems with arithmetic features. The methods of proof of our main result develop estimates for multi-parameter exponential sums, as well as introduce new ideas from the so-called multi-parameter circle method in the context of the geometry of backwards Newton diagrams that are dictated by the shape of the polynomials defining our ergodic averages.