有理距离与有理角度
Rational Distances with Rational Angles
摘要：
在1946年Erd \H OS要求最大数量的单位距离，$u(n)$，之间的$n$点在平面上。他展示了$u(n)> n^{1+c/\log\log n}$，并推测这是真正的量级。最著名的上界是由Spencer、Szemerédi和Trotter提出的$u(n)<cn^{4/3}$。我们表明，上界$n^{1+6/\sqrt{\log n}}$持有，如果我们只考虑单位距离与合理的角度，我们的意思是，通过对点的线使一个合理的角度度与x轴。利用Mann的一个代数定理，得到了单位距离图中两个固定顶点之间的路数的一个一致界，给出了当有过多的有理角的单位距离时的一个矛盾.如果我们考虑有理距离而不是单位距离，只要直线上没有三个点，这个界限就成立。一个超线性下界给出，由于Erd \H OS和珀迪。如果我们在一条直线上有最多的$n^α$点，那么我们分别得到具有有理角的有理距离的数量的界限$O(n^{1+α})$或$n^{1+α+6/\sqrt{\log n}}$，这取决于是$α\ge 1/2$还是$α< 1/2$。
英文摘要：
In 1946 Erd\H os asked for the maximum number of unit distances, $u(n)$, among $n$ points in the plane. He showed that $u(n)> n^{1+c/\log\log n}$ and conjectured that this was the true magnitude. The best known upper bound is $u(n)<cn^{4/3}$, due to Spencer, Szemerédi and Trotter. We show that the upper bound $n^{1+6/\sqrt{\log n}}$ holds if we only consider unit distances with rational angle, by which we mean that the line through the pair of points makes a rational angle in degrees with the x-axis. Using an algebraic theorem of Mann we get a uniform bound on the number of paths between two fixed vertices in the unit distance graph, giving a contradiction if there are too many unit distances with rational angle. This bound holds if we consider rational distances instead of unit distances as long as there are no three points on a line. A superlinear lower bound is given, due to Erd\H os and Purdy. If we have at most $n^α$ points on a line then we get the bound $O(n^{1+α})$ or $n^{1+α+6/\sqrt{\log n}}$ for the number of rational distances with rational angle depending on whether $α\ge 1/2$ or $α< 1/2$ respectively.
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