[Submitted on 17 Jun 2025 (v1), last revised 16 Jun 2026 (this version, v3)]

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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, $1 < p < \infty$. 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, $1 < p < \infty$.

Submission history

From: Bastien Lecluse [view email] [via CCSD proxy]
[v1] Tue, 17 Jun 2025 07:52:28 UTC (23 KB)
[v2] Thu, 11 Jun 2026 07:21:00 UTC (24 KB)
[v3] Tue, 16 Jun 2026 07:11:45 UTC (86 KB)

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