论文标题
有限场中的多项式罗斯定理
A Polynomial Roth Theorem for Corners in Finite Fields
论文作者
论文摘要
我们证明了有限场设置中多项式角的Roth型定理。令$ ϕ_1 $和$ ϕ_2 $为两个不同程度的多项式。对于足够大的素数$ p $，任何子集$ a \ subset \ mathbb f_p \ times \ times \ mathbb f_p $带有$ \ lvert a \ rvert a \ rvert> p ^{2 - \ frac1 {16}} $包含三个点$ （y））$。关于$ \ Mathbb f_p $的这些问题的研究由Bourgain和Chang发起。我们的定理适应了Dong，Li和Sawin的论点，尤其依赖于N. Katz确定的深层类型不平等。
We prove a Roth type theorem for polynomial corners in the finite field setting. Let $ϕ_1$ and $ϕ_2$ be two polynomials of distinct degree. For sufficiently large primes $p$, any subset $ A \subset \mathbb F_p \times \mathbb F_p$ with $ \lvert A\rvert > p ^{2 - \frac1{16}} $ contains three points $ (x_1, x_2) , (x_1 + ϕ_1 (y), x_2), (x_1, x_2 + ϕ_2 (y))$. The study of these questions on $ \mathbb F_p$ was started by Bourgain and Chang. Our Theorem adapts the argument of Dong, Li and Sawin, in particular relying upon deep Weil type inequalities established by N. Katz.
