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Improved estimates for polynomial Roth type theorems in finite fields
Journal d'Analyse Mathématique ( IF 1 ) Pub Date : 2020-08-08 , DOI: 10.1007/s11854-020-0113-8
Dong Dong , Xiaochun Li , Will Sawin

We prove that, under certain conditions on the function pair $\varphi_1$ and $\varphi_2$, bilinear average $p^{-1}\sum_{y\in \mathbb{F}_p}f_1(x+\varphi_1(y)) f_2(x+\varphi_2(y))$ along curve $(\varphi_1, \varphi_2)$ satisfies certain decay estimate. As a consequence, Roth type theorems hold in the setting of finite fields. In particular, if $\varphi_1,\varphi_2\in \mathbb{F}_p[X]$ with $\varphi_1(0)=\varphi_2(0)=0$ are linearly independent polynomials, then for any $A\subset \mathbb{F}_p, |A|=\delta p$ with $\delta>c p^{-\frac{1}{12}}$, there are $\gtrsim \delta^3p^2$ triplets $x,x+\varphi_1(y), x+\varphi_2(y)\in A$. 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.

中文翻译:

有限域中多项式 Roth 型定理的改进估计

我们证明,在函数对 $\varphi_1$ 和 $\varphi_2$ 上的特定条件下,双线性平均 $p^{-1}\sum_{y\in\mathbb{F}_p}f_1(x+\varphi_1(y )) f_2(x+\varphi_2(y))$ 沿着曲线 $(\varphi_1, \varphi_2)$ 满足一定的衰减估计。因此,Roth 型定理在有限域的设置中成立。特别地,如果 $\varphi_1,\varphi_2\in \mathbb{F}_p[X]$ with $\varphi_1(0)=\varphi_2(0)=0$ 是线性无关的多项式,那么对于任何 $A\subset \mathbb{F}_p, |A|=\delta p$ 与 $\delta>cp^{-\frac{1}{12}}$,有 $\gtrsim \delta^3p^2$ 三元组 $x ,x+\varphi_1(y), x+\varphi_2(y)\in A$。这扩展了发起此类问题的 Bourgain 和 Chang 的最新结果,并加强了 Peluse 的结果的界限,Peluse 概括了 Bourgain 和 Chang 的工作。
更新日期:2020-08-08
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