On Assouad dimension of products

A Class of Random Cantor Sets

“…Remark 1.2. The above statements (1) and (2) generalize the results of [9] from one dimensional Moran sets to our model, and the statement (5) when N k are bounded generalize the result of [31] from homogeneous Cantor sets to our model. The proof of dim H E ≥ t * is adapted from [9, Theorem 2.1] to our setting, while the method for the proof of dim B E ≤ s 2 is different from that of [9].…”

“…We can also regard the space Ω as a subclass of Moran sets. The dimensional properties of Moran sets have been studied extensively, we refer to [9,19,21,27,31,37] and reference therein. The results of Theorem 1.1 are similar to the dimensional results of one dimensional homogeneous Cantor sets (uniform Cantor sets).…”

“…For Hausdorff, lower box, upper box, and packing dimensions of one dimensional homogeneous Cantor sets, see [9]. For Assouad dimension of one dimensional homogeneous Cantor sets, see [31]. The Figure 3 corresponds to the partial homogeneous Cantor sets of [9].…”

Remarks on Dimensions of Cartesian Product Sets

“…This completes the proof of (10). Now we prove the equality (9). Let µ be the unique Borel probability measure on…”

“…for arbitrary metric spaces E and F , where dim A E denotes the Assouad dimension of E; see J. Luukkainen [6]. Based on a study on the Assouad dimension of uniform Cantor sets, Peng-Wang-Wen [9] proved that the Assouad dimension of the product E × F may attain any of the values permitted by this inequality.…”