TY - JOUR

T1 - Improved estimates for polynomial Roth type theorems in finite fields

AU - Dong, Dong

AU - Li, Xiaochun

AU - Sawin, Will

N1 - Publisher Copyright:
© 2020, The Hebrew University of Jerusalem.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/9

Y1 - 2020/9

N2 - 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.

AB - 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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U2 - 10.1007/s11854-020-0113-8

DO - 10.1007/s11854-020-0113-8

M3 - Article

AN - SCOPUS:85089065314

VL - 141

SP - 689

EP - 705

JO - Journal d'Analyse Mathematique

JF - Journal d'Analyse Mathematique

SN - 0021-7670

IS - 2

ER -