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arXiv · 2507.15498

Higher-Dimensional Moving Averages and Submanifold Genericity

Jiajun Cheng·Reynold Fregoli·Beinuo Guo
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Abstract

We generalize results of Jones and Olsen on multi-parameter moving ergodic averages to measure-preserving actions of $\mathbb R^d$ for $d\geq 1$. In particular, we give necessary and sufficient conditions for the pointwise convergence of averages over families of boxes in $\mathbb R^d$. As an application of our characterization, we show that averages along dilates of "locally flat" submanifolds in $\mathbb R^d$ do not necessarily converge point-wise for bounded measurable functions. This is closely related to the concept of submanifold-genericity recently introduced in \cite{BFK25}.

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Jiajun Cheng, Reynold Fregoli, Beinuo Guo. 2025-07-21. Higher-Dimensional Moving Averages and Submanifold Genericity. https://arxiv.org/abs/2507.15498

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