Attainable forms of intermediate dimensions

Abstract

The intermediate dimensions are a family of dimensions which interpolate between the Hausdorff and box dimensions of sets. We prove a necessary and sufficient condition for a given function $h(\theta )$ to be realized as the intermediate dimensions of a bounded subset of ${\mathbb{R}}^d$. This condition is a straightforward constraint on the Dini derivatives of $h(\theta )$, which we prove is sharp using a homogeneous Moran set construction.

28 pages, 4 figures. v4: Clarify assumptions of Lemma 3.4 and many other typo fixes; results and numbering unchanged

References in corpus (4)

• Attainable forms of intermediate dimensions
• Generalised intermediate dimensions
• Intermediate dimensions of Bedford-McMullen carpets with applications to Lipschitz equivalence
• Intermediate dimensions of infinitely generated attractors

Cited by in corpus (9)

• Attainable forms of intermediate dimensions
• Intermediate dimensions of Bedford-McMullen carpets with applications to Lipschitz equivalence
• Generalised intermediate dimensions
• Intermediate dimensions of infinitely generated attractors
• Assouad type dimensions of infinitely generated self-conformal sets
• Dimensions of popcorn-like pyramid sets
• Interpolating with generalized Assouad dimensions
• Intermediate dimensions of complementary sets
• Regularity of non-autonomous self-similar sets