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arXiv 1205.0107math.COmath.CAmath.NT
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有限域上平面三角形的面积与Beck定理

Areas of triangles and Beck's theorem in planes over finite fields

Alex Iosevich, Misha Rudnev, Yujia Zhai

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摘要：

本文证明了在有限域$\F_q$上，基数为$|E|>q$的平面的任何子集$E$确定不少于$\frac{q-1}{2}$的三角形的不同面积，而且一次可以找到这样的三角形共享一个公共底.它还表明，如果$|E|\geq 64q\log_2 q$，那么有超过$\frac{q}{2}$不同的区域的三角形共享一个共同的顶点。结果如下从有限域版本的贝克定理的大子集的$\F_q^2$，我们证明。如果$|E|\geq 64q\log_2 q$，则存在点$z\in E$，使得至少有$\frac{q}{4}$条直线入射到$z$，每条直线支持在$\frac{|E|}{2q}$和$\frac{2|E|}{q}.$之间的间隔中除$z$之外的$E$的点的数量。我们还讨论了高维的影响，这些结果在最近的事态发展。

英文摘要：

It is shown that any subset $E$ of a plane over a finite field $\F_q$, of cardinality $|E|>q$ determines not less than $\frac{q-1}{2}$ distinct areas of triangles, moreover once can find such triangles sharing a common base. It is also shown that if $|E|\geq 64q\log_2 q$, then there are more than $\frac{q}{2}$ distinct areas of triangles sharing a common vertex. The result follows from a finite field version of the Beck theorem for large subsets of $\F_q^2$ that we prove. If $|E|\geq 64q\log_2 q$, there exists a point $z\in E$, such that there are at least $\frac{q}{4}$ straight lines incident to $z$, each supporting the number of points of $E$ other than $z$ in the interval between $\frac{|E|}{2q}$ and $\frac{2|E|}{q}.$ This is proved by combining combinatorial and Fourier analytic techniques. We also discuss higher-dimensional implications of these results in light of recent developments.

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