Assouad dimensions of complementary sets

TL;DR

This paper investigates the Assouad dimensions of a class of fractal-like sets formed by prescribed complementary intervals, identifying the range of possible dimensions and their endpoints.

Contribution

It characterizes the set of attainable Assouad-type dimensions for complementary sets defined by a decreasing sequence, including endpoint determination.

Findings

The set of attainable Assouad dimensions often forms a closed interval.

Endpoints of the dimension interval are explicitly determined in many cases.

The results provide a comprehensive understanding of the dimensional spectrum for these sets.

Abstract

Given a positive, decreasing sequence $a,$ whose sum is $L$, we consider all the closed subsets of $[0,L]$ such that the lengths of their complementary open intervals are in one to one correspondence with the sequence $a$. The aim of this note is to investigate the possible values that Assouad-type dimensions can attain for this class of sets. In many cases, the set of attainable values is a closed interval whose endpoints we determine.