On the almost everywhere convergence of two-parameter ergodic averages along directional rectangles

2025/06/17 by Bastien Lecluse, Lecluse, Bastien
Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Approximation and Integration #Mathematical Dynamics and Fractals #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2506.14283

openalex publication_date 2025/06/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we study the almost everywhere convergence of sequences of two-parameter ergodic averages over rectangles in the plane. On the one hand, we show that if the rectangles we consider have their sides with slopes in a finitely lacunary set, then the averages converge almost everywhere in all L^p spaces, 1 < p < ∞. On the other hand, given some non-lacunary sets of directions, we construct sequences of rectangles oriented along these directions for which the associated ergodic averages fail to converge almost everywhere in any L^p space, 1 < p < ∞.

Citations

• Almost everywhere convergence for Lebesgue differentiation processes along rectangles
• Kakeya Sets and Directional Maximal Operators in the Plane

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