In $\mathbb{R}$, we have $n$ finite sets, namely $\{A_1,A_2,\dots, A_n\}$. From them, we define a tiling: $$ T := \{x\in \mathbb{R}^n: \forall i \in \{1,2,\dots,n\}, ~x_i\in A_i\} = \prod_{i=1}^nA_i $$ By difinition, there are $\prod_{i=1}^n|A_i|$ elements in $T$. For example, if $n=2$, $A_1 = \{1,2\}$ and $A_2=\{0\}$, then $T = \{(1,0),(2,0)\}$.

Define size of tiling as $\sigma(T) = \sum_{i=1}^n|A_i|$. Unit circle is denoted as $C =\{x\in \mathbb{R}^n: ||x||_2=1\}$. Given a small value $\epsilon$, we say tiling $T$ induces a $\epsilon$-covering of $C$ if $$ C \subseteq \cup_{x\in T}B(x,\epsilon) $$ where $B(x,\epsilon)$ is a ball centered at $x$ with radius $\epsilon$.

Question 1: find $T_{min}$ such that $\sigma(T_{min})$ = $\min \{\sigma(T)$: $T$ induces a $\epsilon$-covering of $C\}$ and $T_{min}$ induces a $\epsilon$-covering of $C$.

Question 2: if it is hard to find an explicit expression of $T_{min}$ or $\sigma(T_{min})$, can we find a good estimation for $\sigma(T_{min})$?