14 HILLEL FURSTENBERG Thus, by induction, we get Hp(A) Hp 0 (A)m, whence log H∗(A) p log p m log H∗ p0 (A) m log p0 = α. Now consider p [p0 m , p0 m+1 ). Note that the small cubes in the partition of a cube Q into pn equal subcubes can be gathered into at most p0 n (possibly overlapping) cubic blocks of (p0 m )n cubes each. Hence, Hp(A) p0 n Hp 0 (A)m in this case and log Hp(A) log p m log Hp 0 (A) m log p0 + n log p0 log p . Thus, lim sup p→∞ log Hp(A) log p α + lim sup p→∞ n log p0 log p = α. Since p0 was arbitrary here, we conclude that lim sup p→∞ log Hp(A) log p inf p1 log Hp(A) log p lim inf p→∞ log Hp(A)∗ log p . Definition. The limit lim p→∞ log H∗(A) p log p whose existence has been proved in Lemma 2 is called the star dimension of A and denoted by dim∗A. Example 1. Let n = 1 and A = {0} { 1 k : k N} [0, 1]. In this case dim A = 0 (because A is countable). To find the M-dim A, note that when we try to cover A by balls of some small radius r 0, every point 1 k with 1 k 1 k+1 = 1 k(k+1) 2r needs an individual ball, so Nr(A) cr− 1 2 for small r 0. On the other hand, the points 1 k with 1 k(k+1) 2r are all contained in the interval [0,C r], which can be covered by Cr− 1 2 intervals of length 2r. Thus, Nr(A) Cr− 1 2 . These estimates imply immediately that M-dim A = 1 2 . Finally, note that if we take any integer p 1 and partition the interval [0, 1 p ] into p equal subintervals, each of them will contain a point in A, so H∗(A) p = p for all p and dim∗ A = 1. We now formulate the first theorem to be proved by ergodic theoretic means. Although this theorem is not hard to prove by standard considerations, it will serve to illustrate the machinery that will play a crucial role in our later discussion. Theorem 1. If A is a non-empty closed subset of Q = [0, 1]n, then dim∗ A = max{dim A : A is a micro-set of A}. Note that, in particular, the theorem claims that the micro-set of maximal dimension does exist. Also, the possibility to pass to the limit in the Hausdorff metric is crucial here. As the last example shows, we cannot restrict ourselves to mini-sets instead of micro-sets: all mini-sets of A = {0} { 1 k : k N} are countable and, thereby, have Hausdorff dimension 0. Note however that in the Hausdorff metric, we have lim λ→+∞ (λA [0, 1]) = [0, 1], so the micro-sets of A include the whole interval [0, 1], which has Hausdorff dimension 1.
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