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

Jean Bourgain, Mariusz Mirek, Elias M. Stein, James Wright

Research output: Contribution to journalArticlepeer-review

Abstract / Description of output

We prove convergence in norm and pointwise almost everywhere on Lp, p∈(1,∞), 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.
Original languageEnglish
JournalForum of Mathematics, Pi
Publication statusAccepted/In press - 9 Jul 2023


Dive into the research topics of 'On a multi-parameter variant of the Bellow-Furstenberg problem'. Together they form a unique fingerprint.

Cite this