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)) 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∈ F_q[X] with ϕ₁(0) = ϕ₂(0) = 0 are linearly independent polynomials, then for any A⊂ F_q, | A| = δq 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.