Nonlinear Roth type theorems in finite fields

TL;DR

This paper develops smoothing estimates for nonlinear convolution operators over prime fields, resulting in quantitative Roth-type theorems that reveal new phenomena distinct from linear cases.

Contribution

It introduces novel smoothing estimates for nonlinear operators in finite fields, advancing the understanding of nonlinear Roth type theorems.

Findings

Quantitative nonlinear Roth theorems established

Distinct phenomena from linear cases identified

Smoothing estimates for nonlinear convolution operators derived

Abstract

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.