Intermediate Dimension Spectra

• Intermediate dimension spectra are functions that interpolate between the Hausdorff dimension at θ=0 and box dimensions at θ=1 by controlling cover scale disparity.
• They employ constrained covering sums, mass distribution principles, and capacity formalisms to quantify fractal structure with greater sensitivity than traditional methods.
• Explicit models, projection theorems, and random image analyses demonstrate the method’s analytical power in revealing detailed geometric and scaling properties of fractals.

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 (Falconer et al., 2018, Fraser, 2019, Falconer, 2020).

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

$$\overline{\dim}_{\theta} F = \inf \Big\{ s\ge 0 : \text{ for all }\varepsilon>0\ \exists\,\delta_0>0 \text{ such that for all }0<\delta\le \delta_0,$$

$F$0

Equivalently, for $F$1, one may require

$F$2

or, after relabelling scales,

$F$3

for all sets in the cover (Falconer et al., 2018, Fraser, 2019, Fraser, 6 Feb 2025).

The endpoint identifications are exact: $F$4

$F$5

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 (Falconer et al., 2018, Falconer, 2020).

A useful reformulation employs constrained covering sums. For bounded nonempty $F$6,

$F$7

Then $F$8 and $F$9 are characterized as the unique critical values of

$\theta$0

respectively (Burrell et al., 2019, Falconer, 2021).

2. Structural properties and admissible spectral shapes

The spectrum is monotone in $\theta$1: both $\theta$2 and $\theta$3 are increasing on $\theta$4. More quantitatively, if $\theta$5, then

$\theta$6

and similarly for the upper spectrum. In particular, both spectra are continuous on $\theta$7, but continuity at $\theta$8 may fail (Falconer et al., 2018, Falconer, 2020).

The survey literature also records a stronger geometric constraint: the ratios

$\theta$9

are monotone decreasing on $\theta=0$0. This yields a star-shaped graph with respect to the origin, and excludes arbitrary $\theta=0$1-dependence even before finer classification is invoked (Falconer, 2020).

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 $\theta=0$2 there exists a Borel measure $\theta=0$3 supported on $\theta=0$4 with $\theta=0$5 and

$\theta=0$6

then $\theta=0$7 in the appropriate lower or upper sense (Falconer et al., 2018, Fraser, 2019). 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 $\theta=0$8.

The general realizability problem was settled sharply. If $\theta=0$9, then there exists a non-empty bounded set $\theta=1$0 with

$\theta=1$1

if and only if $\theta=1$2 is non-decreasing, continuous on $\theta=1$3, and satisfies

$\theta=1$4

(Banaji et al., 2021). More generally, if one fixes

$\theta=1$5

then the attainable lower and upper spectra are exactly the pairs $\theta=1$6 in the class $\theta=1$7 satisfying

$\theta=1$8

together with $\theta=1$9 and $F\subseteq \mathbb{R}^n$0 (Banaji et al., 2021). 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 (Tan, 2020). The most basic is the polynomial sequence

$F\subseteq \mathbb{R}^n$1

or equivalently $F\subseteq \mathbb{R}^n$2 up to adding isolated points. Its exact spectrum is

$F\subseteq \mathbb{R}^n$3

Hence

$F\subseteq \mathbb{R}^n$4

and the spectrum is a genuine interpolation from $F\subseteq \mathbb{R}^n$5 to $F\subseteq \mathbb{R}^n$6 (Falconer et al., 2018, Fraser, 2019).

The spectrum can, however, be discontinuous at the Hausdorff endpoint. For

$F\subseteq \mathbb{R}^n$7

one has

$F\subseteq \mathbb{R}^n$8

whereas $F\subseteq \mathbb{R}^n$9 (Falconer, 2020). This is the standard example showing that “interpolation” is not automatically genuine at $0\le \theta\le 1$0.

Representative exact formulas are summarized below.

Set	Exact spectrum	Source
$0\le \theta\le 1$1	$0\le \theta\le 1$2	(Falconer et al., 2018)
$0\le \theta\le 1$3	$0\le \theta\le 1$4 if $0\le \theta\le 1$5, else $0\le \theta\le 1$6	(Tan, 2020)
$0\le \theta\le 1$7	$0\le \theta\le 1$8	(Tan, 2020)
$0\le \theta\le 1$9	$$\underline{\dim}_{\theta} F = \inf \Big\{ s\ge 0 : \text{ for all }\varepsilon>0 \text{ and all }\delta_0>0,\ \exists\,0<\delta\le \delta_0$$0 if $$\underline{\dim}_{\theta} F = \inf \Big\{ s\ge 0 : \text{ for all }\varepsilon>0 \text{ and all }\delta_0>0,\ \exists\,0<\delta\le \delta_0$$1, else $$\underline{\dim}_{\theta} F = \inf \Big\{ s\ge 0 : \text{ for all }\varepsilon>0 \text{ and all }\delta_0>0,\ \exists\,0<\delta\le \delta_0$$2	(Tan, 2020)

These examples exhibit several recurring phenomena. Constant spectra occur when the accumulation is too fast, as for geometric radii or for the regime $$\underline{\dim}_{\theta} F = \inf \Big\{ s\ge 0 : \text{ for all }\varepsilon>0 \text{ and all }\delta_0>0,\ \exists\,0<\delta\le \delta_0$$3 in the concentric-sphere family (Tan, 2020). Nontrivial rational spectra arise when clustering is polynomially slow enough to be visible under the $$\underline{\dim}_{\theta} F = \inf \Big\{ s\ge 0 : \text{ for all }\varepsilon>0 \text{ and all }\delta_0>0,\ \exists\,0<\delta\le \delta_0$$4-restricted covering rule (Tan, 2020). Product constructions generate further shapes: for example, if

$$\underline{\dim}_{\theta} F = \inf \Big\{ s\ge 0 : \text{ for all }\varepsilon>0 \text{ and all }\delta_0>0,\ \exists\,0<\delta\le \delta_0$$5

then

$$\underline{\dim}_{\theta} F = \inf \Big\{ s\ge 0 : \text{ for all }\varepsilon>0 \text{ and all }\delta_0>0,\ \exists\,0<\delta\le \delta_0$$6

while $$\underline{\dim}_{\theta} F = \inf \Big\{ s\ge 0 : \text{ for all }\varepsilon>0 \text{ and all }\delta_0>0,\ \exists\,0<\delta\le \delta_0$$7, so the spectrum is strictly increasing on $$\underline{\dim}_{\theta} F = \inf \Big\{ s\ge 0 : \text{ for all }\varepsilon>0 \text{ and all }\delta_0>0,\ \exists\,0<\delta\le \delta_0$$8 but discontinuous at $$\underline{\dim}_{\theta} F = \inf \Big\{ s\ge 0 : \text{ for all }\varepsilon>0 \text{ and all }\delta_0>0,\ \exists\,0<\delta\le \delta_0$$9 (Falconer, 2020).

Bedford–McMullen carpets remain a central unresolved class. Exact formulas are not known, but it is known that

$$\text{and a cover }\{U_i\}\text{ of }F \text{ such that } \delta^{1/\theta}\le |U_i|\le \delta \text{ and } \sum_i |U_i|^s\le \varepsilon \Big\},$$0

for these carpets, and the surveys record the open questions whether, in the non-uniform fibres case, one always has

$$\text{and a cover }\{U_i\}\text{ of }F \text{ such that } \delta^{1/\theta}\le |U_i|\le \delta \text{ and } \sum_i |U_i|^s\le \varepsilon \Big\},$$1

and whether $$\text{and a cover }\{U_i\}\text{ of }F \text{ such that } \delta^{1/\theta}\le |U_i|\le \delta \text{ and } \sum_i |U_i|^s\le \varepsilon \Big\},$$2 is strictly increasing, differentiable, or analytic (Fraser, 2019, Falconer et al., 2018).

4. Capacities, profiles, and projection theorems

A major advance was the replacement of the original covering definition by a capacity theory built from kernels

$$\text{and a cover }\{U_i\}\text{ of }F \text{ such that } \delta^{1/\theta}\le |U_i|\le \delta \text{ and } \sum_i |U_i|^s\le \varepsilon \Big\},$$3

For compact $$\text{and a cover }\{U_i\}\text{ of }F \text{ such that } \delta^{1/\theta}\le |U_i|\le \delta \text{ and } \sum_i |U_i|^s\le \varepsilon \Big\},$$4, the associated capacities are

$$\text{and a cover }\{U_i\}\text{ of }F \text{ such that } \delta^{1/\theta}\le |U_i|\le \delta \text{ and } \sum_i |U_i|^s\le \varepsilon \Big\},$$5

and the lower and upper intermediate dimension profiles $$\text{and a cover }\{U_i\}\text{ of }F \text{ such that } \delta^{1/\theta}\le |U_i|\le \delta \text{ and } \sum_i |U_i|^s\le \varepsilon \Big\},$$6 and $$\text{and a cover }\{U_i\}\text{ of }F \text{ such that } \delta^{1/\theta}\le |U_i|\le \delta \text{ and } \sum_i |U_i|^s\le \varepsilon \Big\},$$7 are defined as the unique critical values satisfying

$$\text{and a cover }\{U_i\}\text{ of }F \text{ such that } \delta^{1/\theta}\le |U_i|\le \delta \text{ and } \sum_i |U_i|^s\le \varepsilon \Big\},$$8

(Burrell et al., 2019, Fraser, 6 Feb 2025).

Ordinary intermediate dimensions are exactly the $$\text{and a cover }\{U_i\}\text{ of }F \text{ such that } \delta^{1/\theta}\le |U_i|\le \delta \text{ and } \sum_i |U_i|^s\le \varepsilon \Big\},$$9 case: $$\overline{\dim}_{\theta} F = \inf \Big\{ s\ge 0 : \text{ for all }\varepsilon>0\ \exists\,\delta_0>0 \text{ such that for all }0<\delta\le \delta_0,$$0 The capacity-covering equivalence is quantitative: $$\overline{\dim}_{\theta} F = \inf \Big\{ s\ge 0 : \text{ for all }\varepsilon>0\ \exists\,\delta_0>0 \text{ such that for all }0<\delta\le \delta_0,$$1 which shows that capacities are not merely auxiliary but encode the same asymptotics as constrained covers (Burrell et al., 2019).

The profile formalism yields a full Marstrand–Mattila theorem for intermediate dimensions. If $$\overline{\dim}_{\theta} F = \inf \Big\{ s\ge 0 : \text{ for all }\varepsilon>0\ \exists\,\delta_0>0 \text{ such that for all }0<\delta\le \delta_0,$$2 is bounded, $$\overline{\dim}_{\theta} F = \inf \Big\{ s\ge 0 : \text{ for all }\varepsilon>0\ \exists\,\delta_0>0 \text{ such that for all }0<\delta\le \delta_0,$$3, and $$\overline{\dim}_{\theta} F = \inf \Big\{ s\ge 0 : \text{ for all }\varepsilon>0\ \exists\,\delta_0>0 \text{ such that for all }0<\delta\le \delta_0,$$4, then for all $$\overline{\dim}_{\theta} F = \inf \Big\{ s\ge 0 : \text{ for all }\varepsilon>0\ \exists\,\delta_0>0 \text{ such that for all }0<\delta\le \delta_0,$$5,

$$\overline{\dim}_{\theta} F = \inf \Big\{ s\ge 0 : \text{ for all }\varepsilon>0\ \exists\,\delta_0>0 \text{ such that for all }0<\delta\le \delta_0,$$6

and for $$\overline{\dim}_{\theta} F = \inf \Big\{ s\ge 0 : \text{ for all }\varepsilon>0\ \exists\,\delta_0>0 \text{ such that for all }0<\delta\le \delta_0,$$7-almost all $$\overline{\dim}_{\theta} F = \inf \Big\{ s\ge 0 : \text{ for all }\varepsilon>0\ \exists\,\delta_0>0 \text{ such that for all }0<\delta\le \delta_0,$$8,

$$\overline{\dim}_{\theta} F = \inf \Big\{ s\ge 0 : \text{ for all }\varepsilon>0\ \exists\,\delta_0>0 \text{ such that for all }0<\delta\le \delta_0,$$9

(Burrell et al., 2019, Fraser, 6 Feb 2025). This identifies the intermediate dimension profiles as the deterministic almost-sure values of projected spectra.

The projection theory also produces an unexpected consequence. If $F$00 is bounded and $F$01 is continuous at $F$02, then

$F$03

if and only if

$F$04

A similar statement holds for upper box dimension (Fraser, 6 Feb 2025). 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 $F$05.

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$06

and let $F$07 be index-$F$08 fractional Brownian motion. Then almost surely, for all $F$09,

$F$10

In particular,

$F$11

This spectrum is strictly smaller than the direct Hölder-image bound

$F$12

so the exact random image is dimensionally sparser, in the intermediate-dimension sense, than a naïve Hölder estimate predicts (Falconer, 2021).

The proof combines a scale-sensitive upper cover with an energy lower bound using the kernels

$F$13

and a comparison kernel

$F$14

adapted to Gaussian increments (Falconer, 2021). 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: $F$15 where $F$16 and $F$17 are the critical values of upper and lower $F$18-pressures defined from cut sets and derivative sums. In the same framework,

$F$19

under explicit scale-regularity and OSC hypotheses (Miao et al., 28 Aug 2025). 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 $F$20-dimensions. These are defined by local covering estimates under the scale-depth restriction

$F$21

recover Assouad dimension when $F$22, recover the $F$23-Assouad spectrum when $F$24, recover quasi-Assouad dimension as $F$25, and recover box dimension when $F$26. In this setting one can realize entire continuous decreasing spectra, and for suitable sets the interval

$F$27

is fully attained by $F$28 as $F$29 (García et al., 2019). This is a different interpolation axis from the Hausdorff–box intermediate dimensions, which constrain global cover comparability rather than local scale depth (Fraser, 2019).

A second distinct usage occurs in the Beurling-dimension theory of spectra of singular spectral measures. For Moran spectral measures,

$F$30

and for each such $F$31 there are continuum many spectra (Li et al., 2023). For a planar self-affine spectral measure generated by

$F$32

one has, for every

$F$33

a spectrum $F$34 with

$F$35

(Chen et al., 22 Oct 2025). For Sierpiński-type spectral measures, every

$F$36

is realized by uncountably many spectra (Li et al., 2023). 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 $F$37 determined by poles of $F$38, while the running spectral dimension $F$39 is a scale-dependent quantity; the paper on quantum spheres and $F$40-Minkowski stresses that these are complementary but distinct notions (Eckstein et al., 2020). In algorithmic dimension theory, the spectrum of a line is

$F$41

and may contain an interval under the hypothesis $F$42 (Lutz et al., 2017). In infinite CIFS theory, the dimension spectrum is

$F$43

the set of subsystem Hausdorff dimensions, which can be compact and perfect yet have Hausdorff dimension zero (Das et al., 2019). 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 $F$44, Frostman and capacity formalisms, a complete realizability theorem, exact projection formulas, and a growing list of explicit examples (Falconer et al., 2018, Banaji et al., 2021, Burrell et al., 2019). 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 (Fraser, 2019).