论文标题
一些孤独的跑步者
A few more Lonely Runners
论文作者
论文摘要
JörgM。Wills提出的寂寞跑步者的猜想，因此由Luis Goddyn提出了命名，自从1967年首次受到构思以来一直是一个有趣的对象：鉴于积极的整数$ k $和$ k $和$ n_1，n_1，n_2，n_2，\ ldots，n_k $，N_K $最初是$ t $ t $ t $ t $ t $ t $ t $ n__ $ \ frac {1} {k+1} $，$ \ forall ~~ 1 \ leq j \ leq k $。在最近的一篇文章贝克（Beck）中，亨特登（Hosten）和舍村（Schymura）描述了孤独的跑步者多面体，并提供了一种多面体方法来识别孤独的跑步者实例的家庭。我们重新审视了孤独的跑步者多面体，并突出了一些满足猜想的新家庭。此外，我们放松了孤独的跑步者多面体中整数点的充分性，以证明猜想。具体而言，我们建议表明孤独的跑步者多面体中整数晶格的某些超级晶格的晶格的存在足够。
Lonely Runner Conjecture, proposed by Jörg M. Wills and so nomenclatured by Luis Goddyn, has been an object of interest since it was first conceived in 1967 : Given positive integers $k$ and $n_1,n_2,\ldots,n_k$ there exists a positive real number $t$ such that the distance of $t\cdot n_j$ to the nearest integer is at least $\frac{1}{k+1}$, $\forall~~1\leq j\leq k$. In a recent article Beck, Hosten and Schymura described the Lonely Runner polyhedron and provided a polyhedral approach to identifying families of lonely runner instances. We revisit the Lonely Runner polyhedron and highlight some new families of instances satisfying the conjecture. In addition, we relax the sufficiency of existence of an integer point in the Lonely Runner polyhedron to prove the conjecture. Specifically, we propose that it suffices to show the existence of a lattice point of certain superlattices of the integer lattice in the Lonely Runner polyhedron.
