Let $A$ and $B$ be fractals with box dimension of $x$ and $y$ respectively. Then prove:

Question

Let $A$ and $B$ be fractals with box dimension of $x$ and $y$ respectively. Then prove that the Cartesian product $A \times B$ has box dimension $x+y$.

Any hints to start out?

(note that box dimension is sometimes called Minkowski dimension)

Answer 1

I don't exactly know what definition you are using, but this is probably easiest to see with the definition $$ \dim A = \lim_{r\to 0} \frac{\log N_A(r)}{-\log r} $$ where $N_A(r)$ is the number of $r$-mesh cubes (i.e., cubes with side length $r$ whose vertices have coordinates which are multiples of $r$) intersecting $A$. It is straightforward to show that $$N_{A \times B} (r) = N_A(r) N_B(r)$$ for all sets $A,B$, and all $r>0$. This means that $$ \log N_{A \times B}(r) = \log N_A(r) + \log N_B(r). $$ Dividing by $-\log r$ and letting $r\to 0$ you get the claim.