On the almost everywhere convergence of two-parameter ergodic averages along directional rectangles
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908410668122112 |
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| author | Lecluse, Bastien |
| author_facet | Lecluse, Bastien |
| contents | 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, $1 < p \leq \infty$. 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, $1 < p < \infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_14283 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the almost everywhere convergence of two-parameter ergodic averages along directional rectangles Lecluse, Bastien Classical Analysis and ODEs 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, $1 < p \leq \infty$. 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, $1 < p < \infty$. |
| title | On the almost everywhere convergence of two-parameter ergodic averages along directional rectangles |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2506.14283 |