Box-Counting Dimension of the Product of a Cantor Set and a Line Segment

From the observation N(1/3n) = 3n⋅2n, we can compute the box-counting dimension.

db = limn→∞Log(N((1/3)n)) / Log(1/((1/3)n)) = limn→∞Log(3n⋅2n) / Log(3n) = limn→∞(Log(3n) + Log(2n)) / (n⋅Log(3)) = limn→∞(n⋅Log(3) + n⋅Log(2)) / (n⋅Log(3)) = 1 + Log(2)/Log(3)

Note this is the sum of the box-counting dimension of the line segment and the Cantor set.

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