Almost sure Assouad-like dimensions of complementary sets

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 Φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varPhi $$\end{document}-dimension of these sets given a natural model of randomness. The Φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varPhi $$\end{document}-dimensions are intermediate Assouad-like dimensions which include the Assouad and quasi-Assouad dimensions as special cases. The answers depend on the size of Φ,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varPhi ,$$\end{document} 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.