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

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.