非阿基米德局部體上的點線重合界

Abstract

對於任意無有限子環的體，我們首先建立一個精確的 Bourgain-Katz-Tao 點線 重合上界。再者，在非阿基米德局部體上，我們給出一個強加乘集界。最後，結合前面兩種結果，我們證明一個非阿基米德局部體平面上的點線重合上界。

除此之外，應用我們證明的點線重合界，能夠探討非阿基米德局部體上的相異距離問題與擴展者問題。

First, we establish an explicit upper bound for the Bourgain-Katz-Tao’s point-line incidence theorem over fields without any finite subrings. Second, we obtain a stronger sum-product bound over non-archimedean local fields. Furthermore, by combining the previous two results, we prove an upper bound for point-line incidence over non-archimedean local fields.

As an application, we use our incidence bounds to study the distinct distance problem and the expander problems over non-archimedean local fields.