how to understand the epsilon-nets are constructed in the followsing set about the $n$-dimentisonal vector?

I am reading an article about constructing nets on a set, but I do not fully understand how the epsilon-nets are constructed. The general idea is to partition the size of the coordinates of a vector and then consider its projection on a cube to construct the nets. However, I am not entirely clear on why this construction works. Below, I will provide specific mathematical definitions and notation, which may appear somewhat messy...

Let $n>d>1$ be large enough integer. Define $n_0=\frac{n\log d}{d}$ and $p=\sqrt{\frac{d}{\log d}}$. Assume that $n_0>p$ (i.e., $1<d<cn^{2/5}\log^{3/5}n$).

For a vector $x\in \mathbb{R}^n$, we fix a permutation $\sigma=\sigma_x$ of $[n]:=\{1,2,\dots, n\}$ such that $x_j^*=|x_{\sigma(j)}|$ for $1\le j\le n$, where we assume that $x_1^*\ge x_2^*\ge \dots \ge x_n^*$.

Define for $i=0, 1,2$,$$T_{0,i}:=\{x\in R^n: x\notin \cup_{j=0}^{i-1} T_{0,j}, x^*_{p^i}>dx^*_{p^{i+1}}\}$$and$$T_0:=\cup_{i=0}^2 T_{0,i}.$$

Fix $I_0\subset [n]$ with $|I_0|=p^3$, we construct a $d^{-3/2}$-net $N_{I_0}$ in the set$$T_{I_0}:=\{x\in (T_0)^c: \sigma_x([p^3])=I_0, x_{p^3}^*\le 1, x_{p^3+1}^*=0\}$$

Assuem that $x^*_{p^3}\le 1$. Since for $x\in (T_0)^c$, then we have$$x_1^*\le dx_p^*\le \dots \le d^3 x_{p^3}^*\le d^3$$

Clearly, the nets $N_{I_0}$ for various $I_0$ can be related by appropriate permutations, so without loss of generality we can assume that $I_0=[p^3]$.

Now, we construct a partition of $I_0$ as follows. Let $$ I_1=[p], I_2=[p^2]\setminus [p]=\{p+1,\dots, p^2\}, \dots, I_3:=[p^3]\setminus [p^2].$$

Then the sets $I_1, \dots, I_3$ form a partition of $I_0$. Now, we consider the set$$T^*:=\{x\in T_{I_0}: \sigma_x(I_j)=I_j, j=1,\dots, 3\}$$and construct a $d^{-3/2}$-nets $N^*$ in $T^*$ as follows.

For every $x\in T^*$, one has $\|P_{I_j}x\|_\infty \le d^{2+2-j}=:b$ for every $1\le j\le 3$, where $P_I$ denotes the coordinate projection onto $\mathbb{R}^I$. Set$$N^*:=N_1\bigoplus N_2 \bigoplus N_3$$where $N_j$ is a $d^{-3/2}$-net in the $\ell_\infty$ metric of cardinality at most $(3bd^{3/2})^{2|I_j|}$ in the coordinate projection of the cube $P_{I_j}(b B_\infty ^n)$, where $B_\infty ^n:=\{x\in \mathbb{R}^n: \|x\|_\infty\le 1\}$.

My question: how to understand that nets $N^*$ and why the cardinality at most $(3bd^{3/2})^{2|I_j|}$?