Improved estimates for polynomial Roth type theorems in finite fields

We prove that, under certain conditions on the function pair ϕ1 and ϕ2, the bilinear average q−1∑y∈Fqf1(x+φ2(y))f2(x+φ2(y))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${q^{- 1}}\sum\nolimits_{y \in {\mathbb{F}_q}} {{f_1}\left({x + {\varphi _2}\left(y \right)} \right){f_2}\left({x + {\varphi _2}\left(y \right)} \right)} $$\end{document} along the curve (ϕ1, ϕ2) satisfies a certain decay estimate. As a consequence, Roth type theorems hold in the setting of finite fields. In particular, if φ1,φ2∈Fq[X]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varphi _1},{\varphi _2} \in {\mathbb{F}_q}\left[X \right]$$\end{document} with ϕ1(0) = ϕ2(0) = 0 are linearly independent polynomials, then for any A⊂Fq,|A|=δq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A \subset {\mathbb{F}_q},\left| A \right| = \delta q$$\end{document} with δ > cq−1/12, there are ≳ δ3q2 triplets x, x+ϕ1(y), x + ϕ2(y) ∈ 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.