Rational Angle Sets and Tight t-Designs

Rational Distances with Rational Angles

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 f…

On the number of perfect triangles with a fixed angle

Richard Guy asked the following question: can we find a triangle with rational sides, medians, and area? Such a triangle is called a \emph{perfect triangle} and no example has been found to date. It is widely believed that such a triangle does not exist. Here we use the setup of Solymosi and de Zeeuw about rational distance sets contained in an algebraic curve, to show that for any angle $0<θ< π$, the number of perfect triangles with an angle $θ$ is finite. A \emph{rational median set} $S$ is a set of points in the plane such that for every three non collinear points $p_1,p_2,p_3$ in $S$ all m…

球面全等五边形铺砌 V：边组合 $a^{4}b$ 与有理角

Tilings of the sphere by congruent pentagons V: Edge combination $a^{4}b$ with rational angles

本文分类了具有边组合 $a^4b$ 和有理角的球面全等五边形边对边铺砌，得到对称与非对称铺砌族及退化情形，并附三维图总结全部结果。

具有有理特征步长的框架与半标准杨表之间的联系

Connections Between Frames with Rational Eigensteps and Semistandard Young Tableaux

本研究建立了有理特征步长框架与半标准杨表的对应关系，提出基于杨表的框架特征步长构造新方法，并用组合技术实现框架补的对应，为框架分析提供组合学新视角。

Macdonald's solid-angle sum for real dilations of rational polygons

The solid-angle sum $A_{\mathcal{P}} (t)$ of a rational polytope ${\mathcal{P}} \subset \mathbb{R}^d$, with $t \in \mathbb{Z}$ was first investigated by I.G. Macdonald. Using our Fourier-analytic methods, we are able to establish an explicit formula for $A_{\mathcal{P}} (t)$, for any real dilation $t$ and any rational polygon ${\mathcal{P}} \subset \mathbb{R}^2$. Our formulation sheds additional light on previous results, for lattice-point enumerating functions of triangles, which are usually confined to the case of integer dilations. Our approach differs from that of Hardy and Littlewood in 1…

全等四边形对球面的铺砌 II：具有有理角的边组合 $a^3 b$

Tilings of the sphere by congruent quadrilaterals II: edge combination $a^3 b$ with rational angles

本文利用三角 Diophantine 方程完整分类所有角为有理度数的 $a^3b$ 全等四边形球面临边铺砌，给出12个孤立情形、3个无限序列及4种例外铺砌，并发现非临边三角形铺砌。