Improved estimates for polynomial Roth type theorems in finite fields

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+\varphi_1(y)) f_2(x+\varphi_2(y))$alongcurve$(\varphi_1, \varphi_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}}$,thereare$\gtrsim \delta^3p^2$triplets$x,x+\varphi_1(y), x+\varphi_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.