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 2ε 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 2ε 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 2ε k 3k − ∞ k=1 2ε 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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