2. B. Box-Counting Dimension
Box-Counting Dimension of the Product of a Cantor Set and a Line Segment
From the observation
N(1/3
n
) = 3
n
⋅2
n
, we can compute the box-counting dimension.
d
b
=
lim
n→∞
Log(N((1/3)
n
)) / Log(1/((1/3)
n
))
=
lim
n→∞
Log(3
n
⋅2
n
) / Log(3
n
)
=
lim
n→∞
(Log(3
n
) + Log(2
n
)) / (n⋅Log(3))
=
lim
n→∞
(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.
Return to
Box-Counting Dimension of the Product of a Cantor Set and a Line Segment
.