---
title: Intermediate Dimension Spectra
url: https://www.emergentmind.com/topics/intermediate-dimension-spectra
type: topic
---

# Intermediate Dimension Spectra

Intermediate dimension spectra are the functions
\[
\theta \mapsto \underline{\dim}_{\theta}F
\qquad\text{and}\qquad
\theta \mapsto \overline{\dim}_{\theta}F,
\qquad \theta\in[0,1],
\]
associated to a bounded set \(F\), where \(\theta\) controls the allowed disparity of diameters in admissible covers. They were introduced to interpolate between Hausdorff dimension at \(\theta=0\) and box dimensions at \(\theta=1\), and have subsequently been treated as an interpolation function carrying more information than the endpoint dimensions alone [1811.06493] [1905.11274] [2011.04363].

## 1. Definitions and endpoint interpolation

For a bounded set \(F\subseteq \mathbb{R}^n\) and \(0\le \theta\le 1\), the lower and upper intermediate dimensions are defined by restricting covers so that all diameters lie in a controlled scale window. In one standard form,
\[
\underline{\dim}_{\theta} F = \inf \Big\{ s\ge 0 : \text{ for all }\varepsilon>0 \text{ and all }\delta_0>0,\ \exists\,0<\delta\le \delta_0
\]
\[
\text{and a cover }\{U_i\}\text{ of }F \text{ such that } \delta^{1/\theta}\le |U_i|\le \delta \text{ and } \sum_i |U_i|^s\le \varepsilon \Big\},
\]
and
\[
\overline{\dim}_{\theta} F = \inf \Big\{ s\ge 0 : \text{ for all }\varepsilon>0\ \exists\,\delta_0>0 \text{ such that for all }0<\delta\le \delta_0,
\]
\[
\text{there is a cover }\{U_i\}\text{ of }F \text{ such that } \delta^{1/\theta}\le |U_i|\le \delta \text{ and } \sum_i |U_i|^s\le \varepsilon \Big\}.
\]
Equivalently, for \(0<\theta\le 1\), one may require
\[
|U_i|\le |U_j|^\theta \qquad \text{for all }i,j,
\]
or, after relabelling scales,
\[
r\le |U_i|\le r^\theta
\]
for all sets in the cover [1811.06493] [1905.11274] [2502.03926].

The endpoint identifications are exact:
\[
\underline{\dim}_{0}F=\overline{\dim}_{0}F=\dim_H F,
\]
\[
\underline{\dim}_{1}F=\underline{\dim}_B F,\qquad \overline{\dim}_{1}F=\overline{\dim}_B F.
\]
Thus intermediate dimensions interpolate from the Hausdorff regime, where covers may use arbitrarily mixed scales, to the box-counting regime, where covers are forced to be essentially single-scale [1811.06493] [2011.04363].

A useful reformulation employs constrained covering sums. For bounded nonempty \(E\subset \mathbb{R}^n\),
\[
S_{r,\theta}^s(E) := \inf\Bigl\{ \sum_i |U_i|^s: \{U_i\}\text{ covers }E,\ r\le |U_i|\le r^\theta \text{ for all }i \Bigr\}.
\]
Then \(\underline{\dim}_\theta E\) and \(\overline{\dim}_\theta E\) are characterized as the unique critical values of
\[
\liminf_{r\to 0}\frac{\log S_{r,\theta}^s(E)}{-\log r}
\quad\text{and}\quad
\limsup_{r\to 0}\frac{\log S_{r,\theta}^s(E)}{-\log r},
\]
respectively [1907.07632] [2108.12306].

## 2. Structural properties and admissible spectral shapes

The spectrum is monotone in \(\theta\): both \(\underline{\dim}_{\theta}F\) and \(\overline{\dim}_{\theta}F\) are increasing on \([0,1]\). More quantitatively, if \(0<\theta<\phi\le 1\), then
\[
\underline{\dim}_\theta F \le \underline{\dim}_\phi F \le \underline{\dim}_\theta F +\Bigl(1-\frac{\theta}{\phi}\Bigr)\bigl(n-\underline{\dim}_\theta F\bigr),
\]
and similarly for the upper spectrum. In particular, both spectra are continuous on \((0,1]\), but continuity at \(\theta=0\) may fail [1811.06493] [2011.04363].

The survey literature also records a stronger geometric constraint: the ratios
\[
\frac{\underline{\dim}_\theta F}{\theta}
\qquad\text{and}\qquad
\frac{\overline{\dim}_\theta F}{\theta}
\]
are monotone decreasing on \((0,1]\). This yields a star-shaped graph with respect to the origin, and excludes arbitrary \(\theta\)-dependence even before finer classification is invoked [2011.04363].

The basic analytic machinery consists of an intermediate-dimension mass distribution principle and an intermediate Frostman lemma. A representative mass distribution statement is: if for all sufficiently small \(r\) there exists a Borel measure \(\mu_r\) supported on \(F\) with \(\mu_r(F)\ge a\) and
\[
\mu_r(U)\le c|U|^s
\qquad\text{for all Borel }U\text{ with }r\le |U|\le r^\theta,
\]
then \(\dim_\theta F\ge s\) in the appropriate lower or upper sense [1811.06493] [1905.11274]. This scale-dependent family of measures is intrinsic to the theory: unlike the classical Hausdorff setting, a single Frostman measure need not suffice because the admissible cover class depends on \(r\).

The general realizability problem was settled sharply. If \(h:[0,1]\to[0,d]\), then there exists a non-empty bounded set \(F\subset\mathbb{R}^d\) with
\[
\dim_\theta F=h(\theta)\qquad (\theta\in[0,1])
\]
if and only if \(h\) is non-decreasing, continuous on \((0,1]\), and satisfies
\[
D^{+}h(\theta)\le \frac{h(\theta)(d-h(\theta))}{d\,\theta}
\qquad\text{for all }\theta\in(0,1)
\]
[2111.14678]. More generally, if one fixes
\[
\dim_L F=\lambda,\qquad \dim_A F=\alpha,
\]
then the attainable lower and upper spectra are exactly the pairs \((\underline h,\overline h)\) in the class \(\mathcal H(\lambda,\alpha)\) satisfying
\[
D^{+}h(\theta)\le \frac{(h(\theta)-\lambda)(\alpha-h(\theta))}{(\alpha-\lambda)\theta},
\]
together with \(\underline h\le \overline h\) and \(\underline h(0)=\overline h(0)\) [2111.14678]. This converts the earlier comparison inequalities into a full classification theorem.

## 3. Explicit formulas and model examples

Exact computations remain comparatively rare, and for that reason a small set of model examples has played a disproportionate role in the subject [2008.10564]. The most basic is the polynomial sequence
\[
F_p=\left\{0,1,\frac{1}{2^p},\frac{1}{3^p},\ldots\right\}
\]
or equivalently \(\{n^{-p}\}\) up to adding isolated points. Its exact spectrum is
\[
\underline{\dim}_{\theta}F_p=\overline{\dim}_{\theta}F_p=\frac{\theta}{p+\theta},
\qquad \theta\in[0,1].
\]
Hence
\[
\dim_H F_p=0,\qquad \dim_B F_p=\frac{1}{p+1},
\]
and the spectrum is a genuine interpolation from \(0\) to \(1/(p+1)\) [1811.06493] [1905.11274].

The spectrum can, however, be discontinuous at the Hausdorff endpoint. For
\[
F_{\log}=\left\{0,\frac1{\log 2},\frac1{\log 3},\frac1{\log 4},\dots\right\},
\]
one has
\[
\underline{\dim}_{\theta}F_{\log}=\overline{\dim}_{\theta}F_{\log}=1
\qquad (\theta\in(0,1]),
\]
whereas \(\dim_HF_{\log}=0\) [2011.04363]. This is the standard example showing that “interpolation” is not automatically genuine at \(\theta=0\).

Representative exact formulas are summarized below.

| Set | Exact spectrum | Source |
|---|---|---|
| \(F_p=\{n^{-p}\}\) | \(\dim_\theta F_p=\dfrac{\theta}{p+\theta}\) | [1811.06493] |
| \(C_p^d=\{x\in\mathbb{R}^d: |x|\in F_p\}\) | \(\dfrac{dp(d-1)+d\theta(1-p(d-1))}{dp+\theta(1-p(d-1))}\) if \(0<p<\frac{1}{d-1}\), else \(d-1\) | [2008.10564] |
| \(T_p=\{(1/t^p,\sin(\pi t)): t\ge 1\}\) | \(1+\dfrac{\theta}{\theta+p}\) | [2008.10564] |
| \(T_{p,q}=\{(1/t^p,t^{-pq}\sin(\pi t)): t\ge 1\}\) | \(\dfrac{p(1+q)+2\theta(1-pq)}{p(1+q)+\theta(1-pq)}\) if \(0<pq<1\), else \(1\) | [2008.10564] |

These examples exhibit several recurring phenomena. Constant spectra occur when the accumulation is too fast, as for geometric radii or for the regime \(p\ge 1/(d-1)\) in the concentric-sphere family [2008.10564]. Nontrivial rational spectra arise when clustering is polynomially slow enough to be visible under the \(\theta\)-restricted covering rule [2008.10564]. Product constructions generate further shapes: for example, if
\[
F=F_1\times F_{\log}\subset\mathbb{R}^2,
\]
then
\[
\underline{\dim}_{\theta}F=\overline{\dim}_{\theta}F=\frac{\theta}{1+\theta}+1
\qquad (\theta\in(0,1]),
\]
while \(\dim_HF=0\), so the spectrum is strictly increasing on \((0,1]\) but discontinuous at \(0\) [2011.04363].

Bedford–McMullen carpets remain a central unresolved class. Exact formulas are not known, but it is known that
\[
\dim_\theta F\to \dim_HF \qquad \text{as }\theta\to 0
\]
for these carpets, and the surveys record the open questions whether, in the non-uniform fibres case, one always has
\[
\dim_\theta F<\dim_BF \quad\text{for all }\theta\in(0,1),
\]
and whether \(\dim_\theta F\) is strictly increasing, differentiable, or analytic [1905.11274] [1811.06493].

## 4. Capacities, profiles, and projection theorems

A major advance was the replacement of the original covering definition by a capacity theory built from kernels
\[
{\phi}_{r,\theta}^{s,k}(x)=
\begin{cases}
1, & 0\le |x|<r,\\[1mm]
\left(\dfrac{r}{|x|}\right)^s, & r\le |x|<r^\theta,\\[2mm]
\dfrac{r^{\theta(k-s)+s}}{|x|^k}, & r^\theta\le |x|.
\end{cases}
\]
For compact \(X\subset\mathbb{R}^d\), the associated capacities are
\[
C_{r,\theta}^{s,k}(X) =
\left(
\inf_{\mu\in\mathcal{M}(X)}
\iint \phi_{r,\theta}^{s,k}(x-y)\,d\mu(x)\,d\mu(y)
\right)^{-1},
\]
and the lower and upper intermediate dimension profiles \(\underline{\dim}_\theta^{\,k}X\) and \(\overline{\dim}_\theta^{\,k}X\) are defined as the unique critical values satisfying
\[
\liminf_{r\to 0}\frac{\log C_{r,\theta}^{s,k}(X)}{-\log r}=s
\quad\text{and}\quad
\limsup_{r\to 0}\frac{\log C_{r,\theta}^{s,k}(X)}{-\log r}=s
\]
[1907.07632] [2502.03926].

Ordinary intermediate dimensions are exactly the \(k=d\) case:
\[
\underline{\dim}_{\theta}X=\underline{\dim}_{\theta}^{\,d}X,
\qquad
\overline{\dim}_{\theta}X=\overline{\dim}_{\theta}^{\,d}X.
\]
The capacity-covering equivalence is quantitative:
\[
r^s C_{r,\theta}^{s,d}(X)\le S_{r,\theta}^s(X)
\le a_d\bigl[\log_2(|X|/r)+1\bigr]\, r^s C_{r,\theta}^{s,d}(X),
\]
which shows that capacities are not merely auxiliary but encode the same asymptotics as constrained covers [1907.07632].

The profile formalism yields a full Marstrand–Mattila theorem for intermediate dimensions. If \(X\subset\mathbb{R}^d\) is bounded, \(1\le k<d\), and \(V\in G(d,k)\), then for all \(V\),
\[
\underline{\dim}_\theta P_V(X)\le \underline{\dim}_\theta^{\,k}X,
\qquad
\overline{\dim}_\theta P_V(X)\le \overline{\dim}_\theta^{\,k}X,
\]
and for \(\gamma_{d,k}\)-almost all \(V\),
\[
\underline{\dim}_\theta P_V(X)= \underline{\dim}_\theta^{\,k}X,
\qquad
\overline{\dim}_\theta P_V(X)= \overline{\dim}_\theta^{\,k}X
\quad \text{for all }\theta\in(0,1]
\]
[1907.07632] [2502.03926]. This identifies the intermediate dimension profiles as the deterministic almost-sure values of projected spectra.

The projection theory also produces an unexpected consequence. If \(X\subset \mathbb{R}^d\) is bounded and \(\underline{\dim}_\theta X\) is continuous at \(0\), then
\[
\underline{\dim}_{\mathrm B} P_V(X)=k
\quad\text{for }\gamma_{d,k}\text{-a.e. }V
\]
if and only if
\[
\dim_{\mathrm H}X\ge k.
\]
A similar statement holds for upper box dimension [2502.03926]. This ties a box-dimensional conclusion about typical projections to a Hausdorff-dimensional hypothesis on the original set through continuity of the intermediate spectrum at \(\theta=0\).

## 5. Random images and non-autonomous extensions

One of the clearest exact random-image spectra is Falconer’s theorem for polynomial sequences under one-dimensional fractional Brownian motion. Let
\[
F_p=\{0,1,1/2^p,1/3^p,\ldots\}\subset \mathbb R,
\]
and let \(B_h:\mathbb R\to\mathbb R\) be index-\(h\) fractional Brownian motion. Then almost surely, for all \(\theta\in[0,1]\),
\[
\underline{\dim}_\theta B_h(F_p)
=
\overline{\dim}_\theta B_h(F_p)
=
\frac{\theta}{ph+\theta}.
\]
In particular,
\[
\dim_B B_h(F_p)=\frac{1}{ph+1}.
\]
This spectrum is strictly smaller than the direct Hölder-image bound
\[
\frac{1}{h}\dim_\theta F_p=\frac{\theta}{h(p+\theta)},
\]
so the exact random image is dimensionally sparser, in the intermediate-dimension sense, than a naïve Hölder estimate predicts [2108.12306].

The proof combines a scale-sensitive upper cover with an energy lower bound using the kernels
\[
\phi_{r,\theta}^s(x)=
\begin{cases}
1,& |x|<r,\\[2mm]
\left(\dfrac{r}{|x|}\right)^s,& r\le |x|<r^\theta,\\[3mm]
\dfrac{r^{\theta(s-1)+1}}{|x|},& |x|\ge r^\theta,
\end{cases}
\]
and a comparison kernel
\[
\psi_{r,\theta}^s(x)=\min\left\{1,\frac{r^{\theta(1-s)+s}}{|x|^h}\right\}
\]
adapted to Gaussian increments [2108.12306]. This is one of the cleanest demonstrations that intermediate spectra can be explicitly computable for nontrivial random images.

A different generalization appears in non-autonomous conformal IFS theory. For non-autonomous conformal sets, the lower and upper intermediate dimensions are given by pressure roots:
\[
\overline{\dim}_\theta E=s^\theta,\qquad \underline{\dim}_\theta E=s_\theta,
\]
where \(s^\theta\) and \(s_\theta\) are the critical values of upper and lower \(\theta\)-pressures defined from cut sets and derivative sums. In the same framework,
\[
\dim_H E=s_*,
\qquad
\overline{\dim}_B E=\dim_P E=s^*
\]
under explicit scale-regularity and OSC hypotheses [2508.20632]. This places intermediate spectra within a thermodynamic formalism parallel to the usual pressure formulas for Hausdorff and box dimensions.

## 6. Related spectra and terminological distinctions

Intermediate dimensions belong to a broader interpolation programme, but they are not the only objects described as spectra. A nearby but distinct theory is the Assouad-side family of \(\Phi\)-dimensions. These are defined by local covering estimates under the scale-depth restriction
\[
0<r\le R^{1+\Phi(R)}<R,
\]
recover Assouad dimension when \(\Phi\equiv 0\), recover the \(\theta\)-Assouad spectrum when \(\Phi(x)=1/\theta-1\), recover quasi-Assouad dimension as \(\Phi\to 0\), and recover box dimension when \(\Phi\to\infty\). In this setting one can realize entire continuous decreasing spectra, and for suitable sets the interval
\[
[\dim_{qA}E,\dim_AE]
\]
is fully attained by \(\overline{\dim}_\Phi E\) as \(\Phi\to 0\) [1903.07155]. This is a different interpolation axis from the Hausdorff–box intermediate dimensions, which constrain global cover comparability rather than local scale depth [1905.11274].

A second distinct usage occurs in the Beurling-dimension theory of spectra of singular spectral measures. For Moran spectral measures,
\[
\forall\, t\in[0,\overline{\dim}_e\mu],\ \exists\ \text{a spectrum }\Lambda_t\text{ with }\dim \Lambda_t=t,
\]
and for each such \(t\) there are continuum many spectra [2302.05868]. For a planar self-affine spectral measure generated by
\[
R=\begin{pmatrix}2&1\\0&2\end{pmatrix},
\qquad
B=\left\{
\begin{pmatrix}0\\0\end{pmatrix},
\begin{pmatrix}1\\0\end{pmatrix}
\right\},
\]
one has, for every
\[
t\in(0,1],\qquad s\in[0,\infty),
\]
a spectrum \(\Lambda_{t,s}\) with
\[
\dim_{Be}(\Lambda_{t,s})=t
\quad\text{and}\quad
D_t^+(\Lambda_{t,s})=s
\]
[2510.19187]. For Sierpiński-type spectral measures, every
\[
t\in\left[0,\frac{\log 3}{\log(3q_2)}\right]
\]
is realized by uncountably many spectra [2303.04047]. These are intermediate-value theorems for Beurling dimensions of frequency sets, not intermediate dimensions of geometric sets in the Falconer–Fraser–Kempton sense.

Other uses of “dimension spectrum” are further removed. In noncommutative geometry and quantum spacetime, the dimension spectrum is a discrete subset of \(\mathbb C\) determined by poles of \(\Gamma(s)\zeta_T(s)\), while the running spectral dimension \(d_S(\sigma)\) is a scale-dependent quantity; the paper on quantum spheres and \(\kappa\)-Minkowski stresses that these are complementary but distinct notions [2005.14210]. In algorithmic dimension theory, the spectrum of a line is
\[
\spec(L_{a,b})=\{\dim(x,ax+b):x\in\mathbb R\},
\]
and may contain an interval under the hypothesis \(\dim(a,b)=\Dim(a,b)\) [1701.04108]. In infinite CIFS theory, the dimension spectrum is
\[
\{\dim_H(\Lambda_A):A\subset E\},
\]
the set of subsystem Hausdorff dimensions, which can be compact and perfect yet have Hausdorff dimension zero [1910.10259]. These usages share a spectral ethos, but they are not intermediate dimensions.

In the strict geometric sense, intermediate dimension spectra now have a mature foundational theory: precise definitions, endpoint identifications, monotonicity, continuity on \((0,1]\), Frostman and capacity formalisms, a complete realizability theorem, exact projection formulas, and a growing list of explicit examples [1811.06493] [2111.14678] [1907.07632]. The main unresolved frontier is no longer the existence of abstract spectra, but explicit computation and regularity in natural fractal classes, most prominently Bedford–McMullen carpets and related self-affine sets [1905.11274].

Source: https://www.emergentmind.com/topics/intermediate-dimension-spectra