Intermediate dimensions 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.

Accepted for publication in Proceedings of the Royal Society of Edinburgh, Section A: Mathematics

References in corpus (5)

• Attainable forms of intermediate dimensions
• Generalised intermediate dimensions
• Projection theorems for intermediate dimensions
• Critical values for Intermediate and Box dimension of projections and other images
• Intermediate dimensions of measures: Interpolating between Hausdorff and Minkowski dimensions