PreprintCC BY-NC-ND 4.0arXiv
Pointwise convergence of bilinear ergodic averages for commuting transformations
Abstract
Let (X,ℬ,μ) be a probability space and let S,Q:X→ X be commuting probability-preserving transformations. Let 1≤ p,q≤∞ satisfy 1/p+1/q≤1. We prove that, for every fixed F∈ Lp(μ) and G∈ Lq(μ), the averages 1/N∑n=1N F(Snx)G(Qnx) converge almost everywhere as N→∞. Our proof introduces a new analytic approach that relies on neither the circle method nor characteristic-factor theory. Exploiting the triangular structure arising from commuting transformations, we combine matrix-energy estimates with Gaussian heat-flow arguments to establish pointwise variation inequalities for suitable smooth averages. We then use even and odd Gaussian kernels to recover the convergence of the original averages.
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BibTeX
@misc{guo2026pointwise,
title = {{Pointwise convergence of bilinear ergodic averages for commuting transformations}},
author = {Zhuowen Guo and Rongzhong Xiao and Shuhao Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26388},
eprint = {2610.26388},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26388},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Pointwise convergence of bilinear ergodic averages for commuting transformations
AU - Zhuowen Guo
AU - Rongzhong Xiao
AU - Shuhao Zhang
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.26388
UR - https://arxiv.org/abs/2610.26388
AB - Let (X,ℬ,μ) be a probability space and let S,Q:X→ X be commuting probability-preserving transformations. Let 1≤ p,q≤∞ satisfy 1/p+1/q≤1. We prove that, for every fixed F∈ L^p(μ) and G∈ L^q(μ), the averages 1/N∑_n=1^N F(S^nx)G(Q^nx) converge almost everywhere as N→∞. Our proof introduces a new analytic approach that relies on neither the circle method nor characteristic-factor theory. Exploiting the triangular structure arising from commuting transformations, we combine matrix-energy estimates with Gaussian heat-flow arguments to establish pointwise variation inequalities for suitable smooth averages. We then use even and odd Gaussian kernels to recover the convergence of the original averages.
ER -