Improved estimates for polynomial Roth type theorems in finite fields
Abstract
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.
 Publication:

arXiv eprints
 Pub Date:
 August 2017
 DOI:
 10.48550/arXiv.1709.00080
 arXiv:
 arXiv:1709.00080
 Bibcode:
 2017arXiv170900080D
 Keywords:

 Mathematics  Number Theory;
 42A38;
 12B99;
 11B05;
 11T23
 EPrint:
 The assumption "having distinct leading terms" is removed in the results and Peluse is acknowledged