Remarks on Dimensions of Cartesian Product Sets

Abstract

Given metric spaces E and F , it is well known thatwhere dimH E, dimP E, dim B E, dimBE denote the Hausdorff, packing, lower box-counting, and upper box-counting dimension of E, respectively. In this note we shall provide examples of compact sets showing that the dimension of the product E ×F may attain any of the values permitted by the above inequalities. The proof will be based on a study on dimension of the product of sets defined by digit restrictions.

A Note on Fractal Measures and Cartesian Product Sets

“…Similar results were proved for s-dimensional Hausdorff measure H s and s-dimensional packing measure P s [2,15,16,31]. The reader can see also [14,25,32,33] for various results on this problem. The aim of this paper is to prove Theorem A and C below.…”

How to avoid a compact set

“…Theorem A is essentially optimal. Using Wei, Wen and Wen [WWW16], we observe that there are compact subsets of R with positive packing dimension all of whose cartesian powers have Hausdorff dimension zero. Thus we can deduce from Theorem B that Theorem A fails when Hausdorff dimension is replaced by packing dimension.…”

“…Given a subset S of the positive integers we let be the lower and upper densities of S, respectively. Let E S be the set of t ∈ [0, 1] such that the kth digit of the binary expansion of t is zero whenever k / ∈ S. Note that E S is compact for every S ⊆ N. The following is a special case of Lemma 1 of [WWW16].…”

A note on the generalized Hausdorff and packing measures of product sets in metric space

“…When q = 0, the measures H q,t µ and P q,t µ do not depend on µ and they will be denoted by H t and P t respectively. The corresponding dimension inequalities for products of these measures are established in [23,31,16], the reader can be referred also to [20,33]. In this case (q = 0), these three inequalities are stated explicitly in [16,14,15,18].…”