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 $a$, the so-called complementary sets. In this paper we determine the almost sure value of the $\mathrm{\Phi }$-dimensions of these sets given a natural model of randomness. The $\mathrm{\Phi }$-dimensions are intermediate Assouad-like dimensions which include the Assouad and quasi-Assouad dimensions as special cases. The answers depend on the size of $\mathrm{\Phi }$, with one size behaving like the Assouad dimension and the other, like the quasi-Assouad dimension.