An optimal generalization of Alon and Füredi's covering result

Given an $n$-cube $\mathcal{Q}^{n} := \{0,1\}^{n}$ in $\mathbb{R}^{n}$, the $k$-th layer $\mathcal{Q}^{n}_{k}$ of $\mathcal{Q}^{n}$ denotes the set of all points in $\mathcal{Q}^{n}$ whose coordinates contain exactly $k$ many ones. In this short note, we consider the following problem: what is the minimum number of hyperplanes in $\mathbb{R}^{n}$ required to cover every point in $\mathcal{Q}^{n} \setminus \mathcal{Q}^{n}_{k}$ at least $t$ times and the points in $\mathcal{Q}^{n}_{k}$ exactly $(t-1)$ times? We prove that the answer to the above question is $\max\left\{ k, n-k\right\}+2t-2$. Note that by putting $k = 0$ and $t=1$, we recover the much celebrated combinatorial geometry result of Alon and F\"{u}redi (European Journal of Combinatorics 1993) where they proved that the minimum number of hyperplanes required to cover every point of $n$-cube except the origin is $n$. We also study a new interesting variant of {\em restricted sumset} problem motivated from the ideas behind the proof of the above result.