Abstract:We obtain new upper bounds on the minimal density $\Theta _{n, \mathcal {K}}$ of lattice coverings of ${\mathbb {R}}^n$ by dilates of a convex body $\mathcal {K}$. We also obtain bounds on the probability (with respect to the natural Haar-Siegel measure on the space of lattices) that a randomly chosen lattice $L$ satisfies $L+\mathcal {K}= {\mathbb {R}}^n$. As a step in the proof, we utilize and strengthen results on the discrete Kakeya problem.

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中文翻译：

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格子覆盖密度的新界限

摘要：我们通过膨胀凸体 $\mathcal { K}$。我们还获得了随机选择的格子 $L$ 满足 $L+\mathcal {K}= {\mathbb {R}}^n$ 的概率的边界（相对于格子空间上的自然 Haar-Siegel 测度） . 作为证明的一步，我们利用并加强了离散 Kakeya 问题的结果。

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