标题： 平面上有限域上的三角形面积和贝克定理 显示英文标题

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

摘要： 显示了平面中有限域 $\F_q$ 的任何子集 $E$，其基数为 $|E|>q$，确定的三角形不同面积不少于 $\frac{q-1}{2}$，而且可以找到共享公共底边的这样的三角形。 还显示了如果 $|E|\geq 64q\log_2 q$，那么存在超过 $\frac{q}{2}$ 个共享公共顶点的不同面积的三角形。 该结果来自于我们证明的适用于 $\F_q^2$ 的大子集的有限域版本的贝克定理。 如果$|E|\geq 64q\log_2 q$，存在一点$z\in E$，使得至少有$\frac{q}{4}$条直线经过$z$，每条直线支持在区间$\frac{|E|}{2q}$和$\frac{2|E|}{q}.$之间的$E$的点数，除了$z$。这是通过组合和傅里叶分析技术结合证明的。 我们还讨论了这些结果在最新发展背景下的高维影响。 显示英文摘要

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.