On almost k-covers of hypercubes

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 ${0, 1}^n \setminus {\vec{0}}$arecoveredatleast$k$times,and$\vec{0}$ is uncovered? The $k=1$ case is the well-known Alon-F"uredi 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 $n$.