Intermediate Dimensions in Fractal Geometry

• Intermediate dimensions are a one-parameter family of fractal dimensions that interpolate between classical Hausdorff and box-counting dimensions using scaled covering constructions.
• The analytical framework employs Dini derivatives in conjunction with Moran-type constructions to provide a sharp criterion for the attainable continuous, non-decreasing spectra.
• These constructions reveal a variety of spectral phenomena and practical implications, bridging geometric analysis with explicit fractal design.

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 (Banaji et al., 2021). 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 (Banaji et al., 2021).

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., (Falconer, 2020)) 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 (Banaji et al., 2021) 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.