Almost sure Assouad-like dimensions of complementary sets

Abstract

Given a non-negative, decreasing sequence a with sum 1, we consider all the closed subsets of [0, 1] such that the lengths of their complementary open intervals are given by the terms of a. These are the so-called complementary sets, or rearrangements of the Cantor set, constructed from a . In this paper we determine the almost sure value of the $$\varPhi $$ -dimension of these sets given a natural model of randomness. The $$\varPhi $$ -dimensions are intermediate Assouad-like dimensions which include the Assouad and quasi-Assouad dimensions as special cases. The answers depend on the size of $$\varPhi ,$$ with one size behaving almost surely like the Assouad dimensions of the associated Cantor set and the other, like the quasi-Assouad dimensions. These results are new even for the Assouad dimensions.