A blueprint for the formalization of norm-variation of multiple ergodic averages for commuting transformations

Abstract

This blueprint serves as a companion to a forthcoming, shorter traditional mathematical paper. The purpose of this blueprint is two-fold: first, it has served as the foundation for a formalization in Lean 4 of these results. This formalization has been completed largely automatically, making essential use of current frontier large language models. Second, it will serve as a resource to readers of the main paper who are interested in further technical details of the proofs. The main result concerns norm-variation estimates for multiple ergodic averages associated with $n\ge 2$ commuting measure preserving transformations, providing a quantitative strengthening of Tao's norm-convergence theorem and answering an open question of Avigad and Rute. At the core of the analysis lies an explicit real-variable estimate for twisted multilinear averages that is closely related to certain singular Brascamp--Lieb inequalities.