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