Let $n>1$ be an integer.
Let $A_n := \{ (x_1,...,x_n)\in \mathbb Z^n : x_i= 0 $ or $1\}$ .
Let $$\mathcal F_n := \{A\subseteq A_n : \text{ for every } a\in A_n , \exists a' \in A \text{ such that } \|a-a'\|\le 1 \},$$
where $\|\cdot\|$ is the euclidean norm . Can we explicitly determine $\inf_{A\in \mathcal F_n} |A| $ as a function of $n$ ? If that is hard, then can we say anything about average/asymptotic order of
$f(n):= \inf_{A\in \mathcal F_n} |A| $ ?