In this paper, we consider the following problem: what is the minimum number of affine hyperplanes in $\mathbb{R}^n$, such that all the vertices of $\{0, 1\}^n \setminus \{\vec{0}\}$ are covered at least $k$ times, and $\vec{0}$ is uncovered? The $k=1$ case is the well-known Alon-Furedi theorem which says a minimum of $n$ affine hyperplanes is required, proved by the Combinatorial Nullstellensatz. We develop an analogue of the Lubell-Yamamoto-Meshalkin inequality for subset sums, and completely solve the fractional version of this problem, which also provides an asymptotic answer to the integral version for fixed $n$ and $k \rightarrow \infty$. We also use a Punctured Combinatorial Nullstellensatz developed by Ball and Serra, to show that a minimum of $n+3$ affine hyperplanes is needed for $k=3$, and pose a conjecture for arbitrary $k$ and large $n$.

中文翻译：

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关于超立方体的几乎 k 覆盖

在本文中，我们考虑以下问题：$\mathbb{R}^n$ 中的最小仿射超平面数是多少，使得 $\{0, 1\}^n \setminus \{\ vec{0}\}$ 至少被覆盖了 $k$ 次，而 $\vec{0}$ 未被覆盖？$k=1$ 的情况是著名的 Alon-Furedi 定理，该定理说至少需要 $n$ 个仿射超平面，由组合 Nullstellensatz 证明。我们为子集和开发了 Lubell-Yamamoto-Meshalkin 不等式的类似物，并完全解决了这个问题的分数版本，这也为固定 $n$ 和 $k \rightarrow \infty$ 的积分版本提供了渐近答案。我们还使用了 Ball 和 Serra 开发的 Punctured Combinatorial Nullstellensatz，以证明 $k=3$ 需要最少 $n+3$ 个仿射超平面，