2. DIMENSION 13 On the other hand, we can use the alternative description of C as the set of all numbers of the form ∑∞ k=1 2εk 3k , εk {0, 1}, (see [KF], page 52, Example 4), and consider the mapping f : [0, 1]\Q [0, 1] defined by f k=1 ε k 2k = ∑∞ k=1 k 3k . Defining the measure μ on [0, 1] \ Q by μ(B) = m(f −1 (B)) where m is the one-dimensional Lebesgue measure, we see that μ(C) = m([0, 1]\Q) = 1. However, if I is any interval of length r [ 1 3l+1 , 1 3l ), then the values of εk, with k l in the representation ∑∞ k=1 k 3k of a number in I are completely determined. Indeed, if ε k = ε k for k k0 l, and ε k0 ε k0 , then k=1 k 3k k=1 k 3k = k=k0 2(ε k ε k ) 3k 2 3k0 kk0 2 3k = 1 3k0 r, so the corresponding points cannot lie in I simultaneously. However, all numbers ∑∞ k=1 εk 2k with fixed ε1,...,εl are contained in an interval of length 2−l, so μ(I) = m(f −1 (I)) 2−l 2r log 2 log 3 . Thus, by Lemma 1, dim C log 2 log 3 . A similar argument shows that the Hausdorff dimension of the Serpinski gasket S equals log 3 log 2 . Note that the converse statement to the statement of Lemma 1 also holds for compact sets A: if dim A = α 0, then for every α α, there exists a measure μ on X satisfying μ(A) 0 and μ(Br(x)) Crα for all x X, r 0. This result is called Frostman’s lemma (see [Mat], pages 112-114) and Chapter 5 of these notes. Thus, at least for compact sets, the Hausdorff dimension is closely related to measures of restricted growth supported on a set. Finally, we will introduce the star dimension. The starting point will be the same as in the case of the box dimension: we will partition a cube into pn subcubes and count the number of subcubes intersecting A [0, 1]n. The crucial difference is that now the cube to be partitioned is not the full cube Q = [0, 1]n, but an arbitrary subcube Q of Q. The formal definition is as follows. For a set A Q define Hp(A) to be the largest number of subcubes that can intersect A in any partition of any cube Q Q into pn equal subcubes. Clearly if A = ∅, we have 1 Hp(A) pn. Lemma 2. The limit lim p→∞ log H∗(A) p log p exists and is a number between 0 and n. Proof. Since 0 log Hp (A) log p n for all p, it suffices to establish the existence of the limit. Fix any p0 1. Let α = log H∗ p0 (A) log p0 . Take a large p of the form p = pm, 0 m N. Note that H∗ pq (A) H∗(A)H∗(A) p q because we can think of the partition of a cube into (pq)n subcubes as of a preliminary partition into pn subcubes followed by the partition of each of those subcubes into qn subsubcubes. At most Hp(A) of the subcubes in the preliminary partition can intersect A, and in each of those that do, at most Hq (A) subsubcubes can intersect A.
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