A Roth theorem in $\mathbb R^2$ and a related ergodic theorem

Danqing He; Xinyu Zhu

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arXiv · 2607.05124

A Roth theorem in $\mathbb R^2$ and a related ergodic theorem

Danqing He·Xinyu Zhu

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Abstract

We prove a quantitative Roth theorem in the plane for the two-dimensional polynomial pattern $(x_1,x_2), (x_1,x_2)+(t_1,t_2), (x_1,x_2)+(t_1^2+t_2^2,t_1^3+t_2^3)$. A pointwise convergence result for the associated polynomial ergodic average is also obtained. A new bilinear Sobolev improving estimate serves as the primary analytic tool, derived from a new sublevel set estimate.

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Danqing He, Xinyu Zhu. 2026-07-06. A Roth theorem in $\mathbb R^2$ and a related ergodic theorem. https://arxiv.org/abs/2607.05124

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