论文标题
圆环上的曲线最多相交
Curves on the torus intersecting at most k times
论文作者
论文摘要
我们表明，圆环上的任何一组独特的同质曲线类别的简单封闭曲线，最多与$ k $ times相交的尺寸$ k + o（\ sqrt {k} \ log k）$。在这项工作之前，Agol的引理，以及用于主要间隙大小的最新情况，暗示了错误术语$ o（k^{21/40}）$，实际上，Riemann假设的假设将此错误术语提高到了我们获得的$ O（\ sqrt {k} {k} {k} \ log log log log k）$。相比之下，我们的方法是基本，组合和几何。
We show that any set of distinct homotopy classes of simple closed curves on the torus that pairwise intersect at most $k$ times has size $k + O(\sqrt{k} \log k)$. Prior to this work, a lemma of Agol, together with the state of the art bounds for the size of prime gaps, implied the error term $O(k^{21/40})$, and in fact the assumption of the Riemann hypothesis improved this error term to the one we obtain $O(\sqrt{k} \log k)$. By contrast, our methods are elementary, combinatorial, and geometric.
