论文标题
在积极特征的固定距离问题上
On the Pinned Distances Problem in Positive Characteristic
论文作者
论文摘要
我们研究了一组$ a \ subset \ mathbb {f}^2 $的ERD \ H OS-FALCONER距离问题，其中$ \ Mathbb {f} $是一个积极特征$ p $的领域。如果$ \ mathbb {f} = \ mathbb {f} _p $和Cardinality $ | a | $超过$ p^{5/4} $，我们证明$ a $确定了可行的$ p $距离的渐近完全比例。对于任何$ \ mathbb {f} $上的$ | a | \ leq p^{4/3} $时，对于小集$ a $，我们证明$ a $ decuns $ a $ decuns $ a $ \ gg | a |^{2/3} $。对于大型和小型组，事实证明的结果实际上是用于固定距离的。
We study the Erd\H os-Falconer distance problem for a set $A\subset \mathbb{F}^2$, where $\mathbb{F}$ is a field of positive characteristic $p$. If $\mathbb{F}=\mathbb{F}_p$ and the cardinality $|A|$ exceeds $p^{5/4}$, we prove that $A$ determines an asymptotically full proportion of the feasible $p$ distances. For small sets $A$, namely when $|A|\leq p^{4/3}$ over any $\mathbb{F}$, we prove that either $A$ determines $\gg|A|^{2/3}$. For both large and small sets, the results proved are in fact for pinned distances.
