CHAPTER 2

Dimension

Since the notion of a tangent plane is absent in the fractal setting, we need an

alternative way to explain why a curve has dimension 1 and a surface has dimension

2. We will now discuss a construction that works in any compact metric space X.

Let A be a closed subset of X. Fix some radius r 0. Let Nr(A) be the

least number of balls of radius r needed to cover A. If A is a rectifiable curve in

Rn,

it is not hard to show that Nr(A) is of order

r−1.

If A is a surface, Nr(A) is

approximately

r−2.

This suggests the idea to define the dimension of an arbitrary

set as the number d for which Nr(A) ≈

r−d.

Definition. The limit lim

r→0+

log Nr (A)

log 1/r

, when it exists, is called the Minkowski

dimension of A and is denoted by M-dimA.

Even if the limit fails to exist, we still can talk about the upper and the lower

limits, which give us the upper and the lower Minkowski dimensions

M-dimA = lim sup

r→0+

log Nr(A)

log

1

r

and M-dimA = lim inf

r→0+

log Nr(A)

log

1

r

correspondingly.

Since Nr(A) is a non-increasing function of r and since log

1

r

changes very

slowly, instead of the (upper or lower) limit along all values of r we can take the

(upper or lower) limit along any sequence rj such that

log 1

rj+1

log 1

rj

→ 1 as j → ∞.

When X = Q is the unit cube in Rn, we can use cubic boxes instead of balls.

More precisely, take some integer p 0 and split Q into pn equal subcubes with

sidelength

1

p

. For A ⊂ Q, let Hp(A) be the number of subcubes that intersect A.

Definition. The box dimension of A is

B-dimA = lim

p→+∞

log Hp(A)

log p

.

Again, in the cases when the limit fails to exist, we may talk about the upper

and the lower box dimensions instead.

Since each cube with sidelength

1

p

is contained in a ball of radius

√

n

p

, say, we

conclude that N

√

n

p

(A) ≤ Hp(A). Since each ball of radius

1

p

can intersect at most

some finite number C(n) cubes of size

1

p

, we have

Hp(A) ≤ C(n) N

1

p

(A).

Taking the logarithm, dividing by log p, and passing to the limit as p → +∞, we

see that the box dimension we just defined coincides with the Minkowski dimension

and the same is true for the upper and lower versions.

11

http://dx.doi.org/10.1090/cbms/120/02