论文标题
有限领域中品种套装的产品
Product of sets on varieties in finite fields
论文作者
论文摘要
令$ v $成为$ \ m athbb {f} _q^d $和$ e \ subset v $中的变体。众所周知，如果通过该来源的任何线包含$ e $的有界数点，则$ | \ prod（e）| = | = | \ {x \ cdot y \ colon x，y \ in e \} | \ gg q $ nesh e \ gg q $ nesh $ nesh $ nesh $时在本文中，我们表明，当$ v $是某些特定维度的抛物面时，屏障$ \ frac {d} {2} $可能会破裂。我们方法中的主要新颖性是将这个问题与一个较低维矢量空间中的距离问题联系起来，从而使我们能够利用该领域的最新发展来获得改进。
Let $V$ be a variety in $\mathbb{F}_q^d$ and $E\subset V$. It is known that if any line passing through the origin contains a bounded number of points from $E$, then $|\prod(E)|=|\{x\cdot y\colon x, y\in E\}|\gg q$ whenever $|E|\gg q^{\frac{d}{2}}$. In this paper, we show that the barrier $\frac{d}{2}$ can be broken when $V$ is a paraboloid in some specific dimensions. The main novelty in our approach is to link this question to the distance problem in one lower dimensional vector space, allowing us to use recent developments in this area to obtain improvements.
