Nonlinear Roth type theorems in finite fields

We obtain smoothing estimates for certain nonlinear convolution operators on prime fields, leading to quantitative nonlinear Roth type theorems. For instance, we produce triplets x, x + y, x + y (2) and x, x + y, x + in proportional subsets of F (p) . Compared with the usual linear setting (i.e. arithmetic progressions), the nonlinear nature of the operators leads to different phenomena, both qualitatively and quantitatively. The methods used are purely analytical.