How do I derive the Minkowski (Box Counting) dimension of the following set?

Question

I have been trying to (rigorously) get the Minkowski's dimension (refer here for a basic definition) of the following:

$(x,y) : \{ y=x, x \in [-2, -1) \ \cup \ (1, 2]\} \ \ \cup \ \ \{ y=0, x \in [-1, 1] \} $.

Answer 1

If your boxes are size $k$, the first two segments can be covered in increments of $k\sqrt 2$ because they are along the diagonal of the square. The last segment is covered in increments of size $k$, so it takes $\frac 4k$ total boxes. The fact that the exponent on the $k$ is $-1$ says the dimension is $1$