The Wikipedia article https://en.wikipedia.org/wiki/Menger_sponge characterizes the successive sets $M_n$ whose intersection is the Menger sponge as follows: $M_0 = [0, 1]^3$ and, for each $n \geq 0$, the set $M_{n+1}$ consists of those points $(x, y, z) \in \mathbb{R}^3$ for which there are integers $i, j, k \in \{0, 1, 2\}$ with $(3x - i, 3y - j, 3z -k) \in M_n$ and at most one of $i, j, k$ is equal to 1.
Are the following analogous descriptions of the Sierpinski carpet $C = \bigcap_{n=0}^{\infty} C_n$ and the Cantor discontinuum $K = \bigcap_{n=0}^{\infty}K_n$ correct?
Sierpinski carpet: $C_0 = [0, 1]^2$ and, for each $n \geq 0$, the set $C_{n+1}$ consists of those points $(x, y) \in \mathbb{R}^2$ for which there are integers $i, j \in \{0, 1, 2\}$ with $(3x - i, 3y - j) \in C_n$ and at most one of $i, j$ is equal to 1.
Cantor discontinuum: $K_0 = [0, 1]$ and, for each $n \geq 0$, the set $K_{n+1}$ consists of those points $x \in \mathbb{R}$ for which there is an integers $i \in \{0, 1, 2\}$ with $3x - i\in K_n$ and $i \neq 1$. (For this dimension, $i = 1$ is excluded.)