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Improved estimates for polynomial Roth type theorems in finite fields

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Abstract

We prove that, under certain conditions on the function pair ϕ1 and ϕ2, the bilinear average \({q^{- 1}}\sum\nolimits_{y \in {\mathbb{F}_q}} {{f_1}\left({x + {\varphi _2}\left(y \right)} \right){f_2}\left({x + {\varphi _2}\left(y \right)} \right)} \) 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 \({\varphi _1},{\varphi _2} \in {\mathbb{F}_q}\left[X \right]\) with ϕ1(0) = ϕ2(0) = 0 are linearly independent polynomials, then for any \(A \subset {\mathbb{F}_q},\left| A \right| = \delta q\) 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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Authors and Affiliations

  1. Center for Scientific Computation and Mathematical Modeling, University of Maryland, College Park, MD, 20742, USA

    Dong Dong

  2. Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA

    Xiaochun Li

  3. Department of Mathematics, Columbia University, New York, NY, 10027, USA

    Will Sawin

Authors
  1. Dong Dong
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  2. Xiaochun Li
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  3. Will Sawin
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Correspondence to Dong Dong.

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Dong, D., Li, X. & Sawin, W. Improved estimates for polynomial Roth type theorems in finite fields. JAMA 141, 689–705 (2020). https://doi.org/10.1007/s11854-020-0113-8

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  • DOI: https://doi.org/10.1007/s11854-020-0113-8

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