# Almost sure Assouad-like Dimensions of Complementary sets

## TL;DR

This paper investigates the almost sure values of intermediate Assouad-like dimensions, called -dimensions, of random complementary sets in [0,1], revealing how these dimensions depend on the size of the function .

## Contribution

It determines the almost sure -dimensions of random complementary sets, connecting these dimensions to the size of , and extends understanding of Assouad-like dimensions.

## Key findings

- Dimensions depend on the size of .
- One size of  behaves like the Assouad dimension.
- Another size behaves like the quasi-Assouad dimension.

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