---
title: Intermediate Dimensions in Fractal Geometry
url: https://www.emergentmind.com/topics/intermediate-dimensions
type: topic
---

# Intermediate Dimensions in Fractal Geometry

Intermediate dimensions constitute a one-parameter family of fractal dimensions interpolating between the classical Hausdorff and box (Minkowski) dimensions of a bounded subset of Euclidean space. By introducing a scale gap in covering constructions, intermediate dimensions capture subtle geometric information that neither endpoint alone reveals. This interpolation function θ ↦ dim_θ E (for θ ∈ [0,1]) is subject to precise analytical constraints; recent work establishes a full characterization of which functions can arise as intermediate dimension spectra [2111.14678]. These results, grounded in Moran-type constructions and analysis via Dini derivatives, demonstrate the flexibility and limits of attainable dimension profiles.

## 1. Definitions: Intermediate Dimensions and Their Properties

Let $E \subset \mathbb{R}^d$ be bounded. For $\theta \in [0,1]$, define a $(\delta, \theta)$-cover as a finite or countable collection $\mathcal U$ of sets covering $E$ such that each $U \in \mathcal U$ satisfies
$$
\delta^{1/\theta} \leq \operatorname{diam} U \leq \delta
$$
(with the convention $\delta^{1/0} = 0$). The $s$-cost of the cover is $C^s(\mathcal U) = \sum_{U \in \mathcal U} (\operatorname{diam} U)^s$.

- **Upper intermediate dimension**:
  $$
  \overline{\dim}_\theta E = \inf \left\{ s \geq 0 : \exists \delta_0 > 0\,\, \forall 0 < \delta \leq \delta_0\,\, \exists (\delta, \theta)\text{–cover}~\mathcal U~\text{with}~C^s(\mathcal U) \leq 1 \right\}
  $$
- **Lower intermediate dimension**:
  $$
  \underline{\dim}_\theta E = \inf \left\{ s \geq 0 : \forall \delta_0 > 0\,\, \exists 0 < \delta \leq \delta_0,\,\, \exists (\delta, \theta)\text{–cover}~\mathcal U~\text{with}~C^s(\mathcal U) \leq 1 \right\}
  $$
If these coincide, write $\dim_\theta E$.

**Special cases**:
- $\underline{\dim}_0 E = \overline{\dim}_0 E = \dim_H E$ (Hausdorff dimension)
- $\underline{\dim}_1 E = \underline{\dim}_B E$, $\overline{\dim}_1 E = \overline{\dim}_B E$ (box-counting dimensions)
- $\theta \mapsto \underline{\dim}_\theta E$, $\overline{\dim}_\theta E$ are non-decreasing and continuous on $(0,1]$

This family bridges the gap between covers of arbitrary size (Hausdorff) and covers with sets of (nearly) uniform size (box).

## 2. Characterization of Attainable Dimension Functions

Given a continuous, non-decreasing function $h\colon [0,1] \to [\lambda, \alpha]$ where $0 \leq \lambda \leq \alpha \leq d$, one seeks conditions for $h$ to be realized as the intermediate dimension spectrum θ↦dim_θ F for some $F \subset \mathbb{R}^d$.

**Sharp criterion via Dini derivatives**:

Define the upper-right Dini derivative
$$
D^+ h(\theta) = \limsup_{h \to 0^+} \frac{h(\theta + h) - h(\theta)}{h}
$$

The function $h$ is *attainable*—i.e., realized as the intermediate dimension function of some bounded set F—if and only if:
- $h$ is non-decreasing;
- $h$ is continuous on $(0,1]$;
- For every $\theta \in (0,1)$,
  $$
  D^+ h(\theta) \leq \frac{(h(\theta) - \lambda)\, (\alpha - h(\theta))}{(\alpha - \lambda)\, \theta}
  $$
When $\lambda = \alpha$, the Dini derivative constraint is vacuous (any continuous, non-decreasing $h$ suffices).

Given two functions $\underline{h}, \overline{h} \in \mathcal{H}(\lambda, \alpha)$, with $\underline{h} \leq \overline{h}$ and $\underline{h}(0) = \overline{h}(0)$, there is $F$ with $\underline{\dim}_\theta F = \underline{h}(\theta)$, $\overline{\dim}_\theta F = \overline{h}(\theta)$ for all θ.

This characterization is sharp and answers open questions on the "possible shapes" of the interpolation curve [2111.14678].

## 3. Constructions Realizing Intermediate Dimension Profiles

The existence part relies on explicit homogeneous Moran set constructions:

- **Step 1: Target function conversion**. Given continuous $h \in \mathcal{H}(0,d)$, define $g(x)$ on $x \in [0, \infty)$ via the correspondence $x = \log \log(1/\delta)$ and $\theta$-parametrization.
- **Step 2: Dini-derivative correspondence**. The Dini-derivative constraint on $h$ converts (via appropriate mean-value/differential inequalities) to $g$-functions with
  $$
  D^+g(x) \in [ -g(x),\, d - g(x) ]
  $$
  which implies integral bounds:
  $$
  d - (d - g(x_0))e^{-(x-x_0)} \geq g(x) \geq g(x_0) e^{-(x-x_0)}
  $$
- **Step 3: Discretization**. Choose contractions $r_1, r_2, \dots \in (0, 1/2]$ so that the stage-dimension $s(\delta) = (n d \log 2)/(-\log \delta)$ at $\delta = r_1\cdots r_n$ satisfies
  $$
  | s(e^{-e^x}) - g(x) | \leq C e^{-x}
  $$
- **Step 4: Homogeneous Moran set construction**. With these $r_j$, the Moran set $C$ satisfies
  $$
  \overline{\dim}_\theta C = \limsup_{x \to \infty} \inf_{y \in [x, x+\log(1/\theta)]} g(y) = h(\theta)
  $$
  Similarly, $\dim_H C = \liminf g(x) = h(0)$. The set's Assouad/lower dimensions attain endpoint values $d$ and $0$.

To match two target functions simultaneously ($\underline{h}, \overline{h}$), one uses inhomogeneous or block-Moran constructions.

This machinery enables the creation of sets whose intermediate spectra are, e.g., strictly convex, strictly concave, linear, or piecewise constant.

## 4. Examples and Spectral Phenomena

A variety of behaviors is attainable for θ ↦ dim_θ F:
- Every non-decreasing Lipschitz $h(\theta)$ with values in $[0,1]$ or $[\lambda,\alpha]$ arises (up to affine rescaling) as the intermediate spectrum of some subset.
- Possible to realize spectra that are piecewise constant, linear, strictly convex/concave, or with prescribed sets of non-differentiability (e.g., arbitrary $G_{\delta\sigma}$ sets of measure zero and even full Hausdorff dimension).
- The intermediate spectrum is always continuous on $(0,1]$, but higher regularity may fail: need not be $C^1$, nor of bounded variation.
- It is possible to construct sets where the set of non-differentiability points is dense and supports full Hausdorff dimension 1.

These findings directly resolve questions posed in Falconer's survey (see, e.g., [2011.04363]) about the general attainable shapes for intermediate dimension curves.

## 5. Relations to Classical Fractal Dimensions

The intermediate dimensions interpolate precisely:
- At θ = 0: dim_0 F recovers Hausdorff dimension.
- At θ = 1: dim_1 F recovers box (Minkowski) dimension.
- θ ↦ dim_θ F is non-decreasing and, if constructed as above, continuous on (0,1], possibly with genuine discontinuity at θ=0 in pathological cases.
- The Dini-derivative constraint is the only nontrivial restriction: except for this, any non-decreasing, continuous shape interpolating between prescribed endpoints can be realized.
- Invariant under bi-Lipschitz maps.

This offers a sharp structural distinction from multifractal spectra or other local dimensions, for which no such explicit parameterized classification is known.

## 6. Analytical and Geometric Implications

The results in [2111.14678] have several implications:
- The dimension spectrum encapsulates strictly more geometric data than any finite set of classical dimensions.
- There is no inherent regularity beyond continuity (on (0,1]) in the interpolation curve—analyticity, convexity, and smoothness are not generic.
- The space of attainable intermediate spectra is maximal: subject only to the endpoint match and Dini-derivative constraint, all interpolating shapes are constructible.
- Methodologically, the connection between Dini-derivative bounds and Moran constructions establishes a bridge between functional-analytic properties of spectra and explicit geometric realizations.

This advances both the constructive and interpolation-theoretic aspects of dimension theory for fractals and highlights the breadth of possible dimension behavior between the Hausdorff and box-counting cases.

Source: https://www.emergentmind.com/topics/intermediate-dimensions