## Abstract

We prove that, under certain conditions on the function pair ϕ_{1} and ϕ_{2}, the bilinear average q−1∑y∈Fqf1(x+φ2(y))f2(x+φ2(y)) along the curve (ϕ_{1}, ϕ_{2}) satisfies a certain decay estimate. As a consequence, Roth type theorems hold in the setting of finite fields. In particular, if φ_{1}, φ_{2}∈ F_{q}[X] with ϕ_{1}(0) = ϕ_{2}(0) = 0 are linearly independent polynomials, then for any A⊂ F_{q}, | A| = δq with δ > cq^{−1}/^{12}, there are ≳ δ^{3}q^{2} triplets x, x+ϕ_{1}(y), x + ϕ_{2}(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.

Original language | English (US) |
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Pages (from-to) | 689-705 |

Number of pages | 17 |

Journal | Journal d'Analyse Mathematique |

Volume | 141 |

Issue number | 2 |

DOIs | |

State | Published - Sep 2020 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Analysis
- General Mathematics