AN INTERMEDIATE VALUE PROPERTY OF FRACTAL DIMENSIONS OF CARTESIAN PRODUCT

Given two metric spaces [Formula: see text], it is well known that, [Formula: see text] where [Formula: see text], [Formula: see text] denote, respectively, the Hausdorff and packing dimension of [Formula: see text]. In this paper, we show that, for any [Formula: see text], there exist [Formula: see text] such that the following equalities hold simultaneously: [Formula: see text] This complete the related results of Wei et al. [C. Wei, S. Y. Wen and Z. X. Wen, Remarks on dimensions of Cartesian product sets, Fractals 24(3) (2016) 1650031].

REMARKS ON DIMENSIONS OF CARTESIAN PRODUCT SETS

Given metric spaces [Formula: see text] and [Formula: see text], it is well known that [Formula: see text] [Formula: see text] and [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] denote the Hausdorff, packing, lower box-counting, and upper box-counting dimension of [Formula: see text], respectively. In this paper, we shall provide examples of compact sets showing that the dimension of the product [Formula: see text] may attain any of the values permitted by the above inequalities. The proof will be based on a study on dimension of products of sets defined by digit restrictions.

Packing dimension, Hausdorff dimension and Cartesian product sets

We show that the dimension adim introduced by R. Kaufman (1987) coincides with the packing dimension Dim, but the dimension aDim introduced by Hu and Taylor [7] is different from the Hausdorff dimension. These results answer questions raised by Hu and Taylor.

On the comparison of measure functions

Suppose that f and g are measure functions and that µf and µg are the corresponding Hausdorff measures. We are interested in how relationships between f and g are reflected in relationships between µf and µg and vice versa. For example, ifRogers ((8), p. 80) has shown that there exists a metric space Ω which has subsets A and B with µf(A) = µg(B) = 0 and, µf(B) = µg(A) = ∞. We are more interested in what happens when the metric space in question is the real line. To get the above result we need to assume that both f and g are starshaped. This is just about a best possible result since if we assume only that f is a power of t the conclusion fails. Further-more we show that there exist measure functions f and g satisfying (1) such that µf and, µg are identical. We also consider related questions: when are, µf and, µg equivalent measures and when are they identical?The proofs depend on the construction in Theorem 3.1 of a Cantor set having pre-scribed measure properties. Although the construction is not difficult it turns out to be quite useful to appeal to the existence of such a set. We illustrate this remark by giving a short proof of a known theorem on cartesian product sets. We also make use of these ideas in section 5 where we discuss some properties of a class of net meastues and give a partial answer to a problem posed by Billingsley.