Homogeneity and non-coincidence of Hausdorff and box dimensions for subsets of Rⁿ

Abstract

Abstract. A class of subsets of R n is constructed that have certain homogeneity and non-coincidence properties with respect to Hausdorff and box dimensions. For each triple (r, s, t) of numbers in the interval (0, n] with r < s < t, a compact set K is constructed so that for any non-empty subset U relatively open in K, we have

Coincidence and noncoincidence of dimensions in compact subsets of $[0,1]$

“…Finally, one could ask whether our set exhibits homogeneity properties similar to those satisfied by the set constructed by Nilsson and Wingren in [12]. In their paper, Nilsson and Wingren show that given any three numbers r, s, t ∈ (0, d] with r < s < t, it is possible to construct a compact subset K of R d with dim H (K ∩ U ) = r, dim B (K ∩ U ) = s and dim B (K ∩ U ) = t for every open set U with K ∩ U = ∅.…”

“…where Σ * 2 denotes the set of all finite binary words. Hence, for each w ∈ Σ * 2 there is a unique positive integer k such that f −1 (k) = w. We are now in a position to give the exact definition of (12) satisfies 5), ( 8), ( 10) and ( 12), and set…”

“…There are many examples in the literature of sets in which one or more of the above inequalities are strict, see, for example, [4], [5], [12], [13], [14], [15], [16]. We particularly draw the reader's attention to the result of Spear [15].…”

“…Let r, s, t, u, v, w ∈ [0, 1] be as in the statement of Theorem 1.1, n = (n k ) k as in the statement of Lemma 3.1, k = (k n ) n as in the statement of Lemma 3.2, j = (j n ) n as in the statement of Lemma 3.3 and a = (a k ) k and b = (b k ) k as in the statement of Lemma 3.4. Let C = C(r, t, n), D = D(u, v, n, k), E = E(w, j) and F = F (s, a, b), as defined in (5), (8), (10) and(12), and set X = C ∪ D ∪ E ∪ F . It follows by Lemmas 3.1, 3.2, 3.3, and 3.4 that the dimensions of C, D, E and F are as follows.…”