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

Danqing He; Xinyu Zhu

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

Classical Analysis and ODEs 2026-07-06 v1

Authors: Danqing He , Xinyu Zhu

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.

Cite

@article{arxiv.2607.05124,
  title  = {A Roth theorem in $\mathbb R^2$ and a related ergodic theorem},
  author = {Danqing He and Xinyu Zhu},
  journal= {arXiv preprint arXiv:2607.05124},
  year   = {2026}
}