Let $K$ be a bounded convex domain in ${\mathbb{R}}^2$ symmetric about the origin. The critical locus of $K$ is defined to be the (non-empty compact) set of lattices $\Lambda$ in ${\mathbb{R}}^2$ of smallest possible covolume such that $\Lambda \cap K=\left\{0\right\}$. These are classical objects in geometry of numbers; yet all previously known examples of critical loci were either finite sets or finite unions of closed curves. In this paper we give a new construction which, in particular, furnishes examples of domains having critical locus of arbitrary Hausdorff dimension between 0 and 1.

中文翻译：

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平面中凸域的临界位点

让 $ķ$ 是...的有界凸域 ${\mathbb{[R}}^{\mathrm{2个}}$关于原点对称的。该关键位点的$ķ$ 被定义为（非空紧凑型）晶格集合 $\Lambda$ 在 ${\mathbb{[R}}^{\mathrm{2个}}$ 最小可能的体积 $\Lambda \cap ķ=\left\{0\right\}$。这些是数字几何中的经典对象。但是，所有以前已知的临界位点示例都是闭合曲线的有限集或有限并集。在本文中，我们给出了一种新的结构，特别是提供了具有Hausdorff维数在0和1之间的任意临界位点的域的示例。