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. Compared with the usual linear setting (i.e. arithmetic progressions), the nonlinear nature of the operators leads to different phenomena, both qualitatively and quantitatively. 
Jean Bourgain and Mei-Chu Chang
math.NT