Intermediate dimensions of complementary sets

TL;DR

This paper investigates the intermediate dimensions of complementary sets formed by rearranging a given sequence, providing explicit calculations of the range of these dimensions and filling a gap in the understanding of their properties.

Contribution

It characterizes the full range of intermediate dimensions for complementary sets and explicitly computes the endpoints of this range under certain conditions.

Findings

Range of $ heta$-intermediate dimensions forms a closed interval

Endpoints of the interval are explicitly computed

Results fill a gap in the literature on dimensional properties of complementary sets

Abstract

Given a positive, non-increasing sequence $a$ with finite sum equal to $1$, we consider the family of all closed subsets of $[0,1]$ whose complementary open intervals have lengths given by a rearrangement of the sequence $a$. We study the full range of possible $\theta$-intermediate dimensions of these sets and, under suitable assumptions on the sequence, we show that this range forms a closed interval, whose endpoints we compute explicitly. This paper fills a gap in the literature concerning the dimensional properties of complementary sets.