Improved estimates for polynomial Roth type theorems in finite fields

Abstract

We prove that, under certain conditions on the function pair ${\phi }_1$ and ${\phi }_2$, bilinear average $p^{-1}{\sum }_{y\in {\mathbb{F}}_p}{f}_1(x+{\phi }_1(y))f_2(x+{\phi }_2(y))$ along curve $({\phi }_1,{\phi }_2)$ satisfies certain decay estimate. As a consequence, Roth type theorems hold in the setting of finite fields. In particular, if ${\phi }_1,{\phi }_2\in {\mathbb{F}}_p[X]$ with ${\phi }_1(0)={\phi }_2(0)=0$ are linearly independent polynomials, then for any $A\subset {\mathbb{F}}_p,|A|=\delta p$ with $\delta >c{p}^{-\frac{1}{12}}$, there are $\gtrsim {\delta }^3{p}^2$ triplets $x,x+{\phi }_1(y),x+{\phi }_2(y)\in A$. This extends a recent result of Bourgain and Chang who initiated this type of problems, and strengthens the bound in a result of Peluse, who generalized Bourgain and Chang's work. The proof uses discrete Fourier analysis and algebraic geometry.

The assumption "having distinct leading terms" is removed in the results and Peluse is acknowledged