Assouad Spectrum in Fractal Geometry

• Assouad spectrum is a one-parameter family of dimensions defined by fixing a scale relation to interpolate between box-counting and quasi-Assouad dimensions.
• Research reveals its rich behavior including strict concavity, phase transitions, and explicit formulas in self-affine and random models.
• Its robustness under bi-Lipschitz transformations and unique projection properties make it a valuable tool for analyzing complex local geometries.

Searching arXiv for recent and foundational papers on the Assouad spectrum. The Assouad spectrum is a one-parameter family of dimensions that constrains the relation between the large and small scales in local covering problems, thereby interpolating between upper box-counting dimension and quasi-Assouad dimension, and in many important settings reflecting how extremal local geometry emerges across relative scale separation. Introduced by Fraser and Yu and further developed in later work, it is defined for bounded sets $F\subset \mathbb{R}^d$ by fixing a parameter $\theta\in(0,1)$ and requiring the outer scale to be $R=r^\theta$, so that the spectrum records the covering complexity of $B(x,r^\theta)\cap F$ at scale $r$ uniformly over $x\in F$ (Fraser et al., 2018). Recent work has shown that the Assouad spectrum is not merely an interpolation device: it can distinguish typical graphs in Banach spaces (Feng et al., 9 Jan 2026), exhibit strictly concave and non-piecewise differentiable behaviour (Fraser et al., 2018), encode fine self-affine structure in Gatzouras–Lalley carpets (Banaji et al., 2024), and fail to satisfy Marstrand-type projection theorems in the same way as the quasi-Assouad dimension (Falconer et al., 27 Jun 2026).

1. Definition and dimensional position

For a bounded non-empty set $F\subset \mathbb{R}^d$, let $N_r(E)$ denote the least number of radius-$r$ balls needed to cover a bounded set $E$. The Assouad dimension is

$\theta\in(0,1)$0

so it probes worst-case local scaling over all locations and all pairs of scales (Fraser et al., 2018).

The Assouad spectrum fixes the scale relation $\theta\in(0,1)$1, equivalently $\theta\in(0,1)$2. In one standard formulation,

$\theta\in(0,1)$3

for $\theta\in(0,1)$4 (Fraser et al., 2018). An equivalent formulation used in other works is

$\theta\in(0,1)$5

which makes the outer scale $\theta\in(0,1)$6 explicit (Feng et al., 9 Jan 2026).

This family sits between box and Assouad-type quantities. For bounded $\theta\in(0,1)$7,

$\theta\in(0,1)$8

and in fact

$\theta\in(0,1)$9

with $R=r^\theta$0 continuous on $R=r^\theta$1 (Feng et al., 9 Jan 2026). As $R=r^\theta$2, the spectrum tends to upper box-counting dimension, while as $R=r^\theta$3 it tends to quasi-Assouad dimension (Fraser et al., 2018). This identifies the Assouad spectrum as a scale-resolved invariant positioned strictly between global one-scale quantities and fully extremal two-scale quantities.

A related notion is the upper Assouad spectrum

$R=r^\theta$4

which a priori allows all $R=r^\theta$5 rather than the single relation $R=r^\theta$6 (Fraser et al., 2018). A fundamental theorem shows that it carries no additional information: $R=r^\theta$7 This result explains why the Fraser–Yu definition is sufficient for recovering the corresponding “upper” data (Fraser et al., 2018).

2. Endpoint behaviour and quasi-Assouad dimension

A central structural fact is that the Assouad spectrum always converges to the quasi-Assouad dimension at the right endpoint. Fraser, Hare, Hare, Troscheit and Yu proved that

$R=r^\theta$8

for all $R=r^\theta$9 (Fraser et al., 2018). This shows that the natural endpoint of the spectrum is not the full Assouad dimension in general, but the quasi-Assouad dimension.

The distinction is substantive. The full Assouad dimension allows arbitrary pairs of scales $B(x,r^\theta)\cap F$0, whereas the spectrum samples only power-law related pairs. This means that scale configurations responsible for $B(x,r^\theta)\cap F$1 may be too sparse or too anisotropic to be seen along the constrained relation $B(x,r^\theta)\cap F$2. The paper “The Assouad spectrum and the quasi-Assouad dimension: a tale of two spectra” explicitly states that the full Assouad dimension can fail to appear as the right-hand limit, while quasi-Assouad always does (Fraser et al., 2018).

This phenomenon is especially transparent in recent work on typical graphs. For the modulus $B(x,r^\theta)\cap F$3, one has

$B(x,r^\theta)\cap F$4

In the associated little $B(x,r^\theta)\cap F$5-space, a typical graph satisfies

$B(x,r^\theta)\cap F$6

while for every $B(x,r^\theta)\cap F$7 and every $B(x,r^\theta)\cap F$8,

$B(x,r^\theta)\cap F$9

(Feng et al., 9 Jan 2026). Thus the spectrum and quasi-Assouad dimension remain at the minimal graph value $r$0, whereas the full Assouad dimension of a typical graph is maximal. The paper interprets this by observing that the “worst scale separation” appears only when $r$1 and $r$2 are allowed to vary independently; tying them by $r$3 suppresses that extreme behaviour (Feng et al., 9 Jan 2026).

This suggests that the spectrum is often more robust than $r$4 with respect to rare local irregularities. A plausible implication is that, in applications where geometric information should ignore exceptional sub-exponential scale interactions, the quasi-Assouad endpoint may be more stable than the full Assouad dimension.

3. Functional and structural properties

The shape of the Assouad spectrum can be much richer than early examples suggested. The 2018 paper (Fraser et al., 2018) established a realisation theorem: if $r$5 is continuous, concave, non-decreasing, satisfies $r$6, and obeys

$r$7

then there exists a compact set $r$8 such that

$r$9

Consequently, the spectrum can be strictly concave, can exhibit phase transitions of any order, need not be piecewise differentiable, and need not be constant in any neighbourhood of $x\in F$0 (Fraser et al., 2018).

This broad realisability contrasts with the behaviour of many classical examples, where the spectrum is piecewise linear or reaches a plateau near $x\in F$1. The same paper records a conjecture that for every set $x\in F$2 there exists $x\in F$3 such that

$x\in F$4

but this remains a conjectural regularity statement (Fraser et al., 2018).

A broader functional framework is provided by intermediate Assouad-like dimensions, indexed by a dimension function $x\in F$5. For a dimension function $x\in F$6, the upper $x\in F$7-dimension is defined by constraining $x\in F$8 rather than fixing a single power-law relation. This recovers the Assouad dimension when $x\in F$9, and the $F\subset \mathbb{R}^d$0-Assouad spectrum when $F\subset \mathbb{R}^d$1 is constant (García et al., 2019). The same paper shows that $F\subset \mathbb{R}^d$2-dimensions interpolate between box, quasi-Assouad, and Assouad dimensions, and that there are central Cantor sets for which the family $F\subset \mathbb{R}^d$3 fills the full interval $F\subset \mathbb{R}^d$4 (García et al., 2019).

This larger viewpoint makes clear that the classical Assouad spectrum is a distinguished one-parameter slice through a more flexible family of local dimensions. A plausible implication is that some phenomena first observed for $F\subset \mathbb{R}^d$5-spectra may be better understood in the $F\subset \mathbb{R}^d$6-framework, particularly when critical scale transitions occur at non-power-law rates.

4. Explicit formulas in self-affine and random settings

For self-affine sets, the Assouad spectrum frequently admits explicit formulas that reflect anisotropy and inhomogeneity. In deterministic Bedford–McMullen carpets, Fraser’s book records that for $F\subset \mathbb{R}^d$7,

$F\subset \mathbb{R}^d$8

and for $F\subset \mathbb{R}^d$9,

$N_r(E)$0

(Fraser, 2020). Thus the spectrum is piecewise affine with a single phase transition.

For Gatzouras–Lalley carpets, the structure is substantially richer. Banaji, Fraser, Kolossváry and Rutar proved that if $N_r(E)$1 is such a carpet, then

$N_r(E)$2

where $N_r(E)$3 is a concave column pressure function obtained as a minimum of finitely many analytic functions and $N_r(E)$4 is its concave conjugate (Banaji et al., 2024). Their corollary gives a piecewise description with four regimes: a small-$N_r(E)$5 box-type region, interior curved regions associated to inhomogeneous columns, linear bridge intervals, and an Assouad plateau for large $N_r(E)$6 (Banaji et al., 2024). This formula yields several phenomena “not previously observed for dynamically invariant sets”, including differentiable nontrivial spectra on the whole interval $N_r(E)$7, strict concavity on open intervals, and phase transitions of arbitrary odd order (Banaji et al., 2024).

Random non-conformal models produce another distinctive profile. For random self-affine Bedford–McMullen carpets, Fraser and Troscheit obtained almost sure formulas for $N_r(E)$8 that are piecewise explicit, with a phase transition at

$N_r(E)$9

and a constant branch equal to the quasi-Assouad dimension for $r$0 (Fraser et al., 2018). In this model, the almost sure Assouad dimension is typically larger and more rigid than the quasi-Assouad dimension, while the spectrum sits strictly between box and quasi-Assouad dimensions on $r$1 (Fraser et al., 2018). This contrasts with the deterministic Bedford–McMullen case, where $r$2, and with some random conformal settings, where quasi-Assouad collapses to box dimension (Fraser et al., 2018).

The same interpolation philosophy has recently been exported to dynamics. The paper “Mean Assouad dimension and spectrum, with applications to infinite dimensional fractals” defines the mean Assouad spectrum for a topological dynamical system $r$3 by replacing local covering numbers with exponential-in-time growth rates over Bowen balls, and proves

$r$4

(Huo et al., 1 Jan 2026). In infinite-dimensional Bedford–McMullen carpet systems, the mean Assouad spectrum is piecewise linear with a single phase transition at $r$5 (Huo et al., 1 Jan 2026). This indicates that the scale-interpolation paradigm underlying the Assouad spectrum extends naturally beyond static fractal sets.

5. Graphs of functions and typical behaviour

Graphs of functions provide one of the most active current arenas for the Assouad spectrum. The general problem is to understand how analytic regularity constraints interact with local covering growth on $r$6.

For $r$7-Hölder functions $r$8, Chrontsios-Garitsis and Tyson proved the upper bound

$r$9

and showed it is sharp by constructing $E$0-Hölder graphs attaining equality for all $E$1 (Garitsis et al., 2023). Their geometric algorithm starts from a graph satisfying both upper and lower Hölder oscillation bounds and modifies it by reflections inside shrinking squares to force large local covering numbers without violating Hölder regularity (Garitsis et al., 2023). They also proved a Sobolev counterpart: if $E$2 is continuous, then

$E$3

and this too is sharp (Garitsis et al., 2023).

The 2026 paper on typical graphs takes this further by moving from individual constructions to Baire-typical behaviour in Banach spaces. Let $E$4 be a concave modulus of continuity and

$E$5

Then for a typical $E$6, if $E$7,

$E$8

In particular, in the little $E$9-Hölder space with $\theta\in(0,1)$00,

$\theta\in(0,1)$01

while $\theta\in(0,1)$02 and $\theta\in(0,1)$03 for a typical graph (Feng et al., 9 Jan 2026).

These results show that the spectrum can capture a gradual transition from box-type restrictions to full local extremality. In little Hölder spaces, the regularity constraint forces

$\theta\in(0,1)$04

yet a typical graph still has full Assouad dimension $\theta\in(0,1)$05 (Feng et al., 9 Jan 2026). The spectrum records the exact scale at which the Hölder constraint ceases to limit local complexity. By contrast, in the log-modified modulus space $\theta\in(0,1)$06, every graph has $\theta\in(0,1)$07 for all $\theta\in(0,1)$08, while a typical graph has $\theta\in(0,1)$09 (Feng et al., 9 Jan 2026). This makes the spectrum an especially sharp diagnostic for distinguishing function spaces whose graphs share the same Assouad dimension.

6. Distortion, projections, and current frontiers

The Assouad spectrum is stable under bi-Lipschitz changes of metric, but its behaviour under weaker or geometric transformations is subtler. For Euclidean quasiconformal maps, the regularized Assouad spectrum satisfies explicit distortion inequalities. If $\theta\in(0,1)$10 is $\theta\in(0,1)$11-quasiconformal and $\theta\in(0,1)$12 is compact, then for every $\theta\in(0,1)$13,

$\theta\in(0,1)$14

$\theta\in(0,1)$15

where $\theta\in(0,1)$16 (Garitsis et al., 2021). In the plane, these estimates simplify using Astala’s sharp higher-integrability exponent and can be used to classify polynomial spirals $\theta\in(0,1)$17: for $\theta\in(0,1)$18, there exists a quasiconformal map $\theta\in(0,1)$19 with $\theta\in(0,1)$20 if and only if $\theta\in(0,1)$21 (Garitsis et al., 2021). The classification depends not on Hausdorff or Assouad dimension alone—both are uninformative here—but on the precise location where the spectrum reaches $\theta\in(0,1)$22.

Projection theory reveals both the power and the limits of the spectrum. A 2026 paper shows that Marstrand’s projection theorem fails for the quasi-Assouad dimension and for the Assouad spectrum: there exist uniformly discrete unbounded planar sets whose projected spectra take different prescribed values on disjoint open sets of directions (Falconer et al., 27 Jun 2026). At the same time, the paper proves almost sure lower bounds using capacity-theoretic dimension profiles and almost sure upper bounds for bounded planar sets via a tube-counting argument (Falconer et al., 27 Jun 2026). Thus the projection theory of the Assouad spectrum resembles that of Assouad dimension in that classical almost sure constancy fails, but it also admits refined profile-based lower bounds analogous to box-dimension projection theory.

There are also measure-theoretic and lower-spectrum analogues. For measures, upper and lower Assouad spectra converge to the corresponding quasi-Assouad dimensions under finite quasi-upper Assouad dimension, and continuity in $\theta\in(0,1)$23 holds under the same hypothesis (Hare et al., 2018). On the lower side, the lower Assouad spectrum converges to the quasi-lower Assouad dimension for uniformly perfect sets in doubling metric spaces, providing an equivalent definition of the latter (Chen et al., 2018). These developments suggest that the spectrum framework is part of a broader family of scale-interpolating dimensions rather than a stand-alone invariant.

Taken together, the literature indicates that the Assouad spectrum is most informative precisely when classical dimensions disagree or when one wants to separate “typical” from “extremal” local geometry. It can be strictly concave (Fraser et al., 2018), piecewise linear (Fraser, 2020), constant while $\theta\in(0,1)$24 is maximal (Feng et al., 9 Jan 2026), or shaped by explicit dual variational principles in self-affine settings (Banaji et al., 2024). This suggests that the Assouad spectrum should be viewed not as a minor variant of $\theta\in(0,1)$25, but as a central multiscale invariant in contemporary fractal geometry.