Quasi-Assouad Dimension Overview

• Quasi-Assouad Dimension is an Assouad-type measure that quantifies maximal local covering growth with a restricted scale gap to ignore sub-exponential effects.
• It is defined as the limit of covering estimates under a controlled scale relationship and is linked to the endpoint of the Assouad spectrum.
• Its applications span deterministic fractal constructions, random models, and measure theory, providing sharper insights into local inhomogeneity in fractal geometry.

The quasi-Assouad dimension is an Assouad-type dimension for bounded metric sets that measures maximal local covering growth under a restriction that the smaller scale is not exponentially smaller than the larger one. Introduced by Lü and Xi, it refines the classical Assouad dimension by suppressing certain extremal short-scale effects while retaining a local, worst-case character; in contemporary dimension theory it is also identified as the right-endpoint limit of the Assouad spectrum introduced by Fraser and Yu (Lü et al., 2014, Fraser et al., 2018).

1. Definition and basic formulation

For a bounded set $F \subset \mathbb{R}^d$, let $N(E,r)$ denote the minimal number of sets of diameter at most $r$ needed to cover $E$. The classical Assouad dimension is

$$\dim_A F = \inf \left\{ s : \exists C>0 \text{ s.t. } \forall 0<r<R<1,\ \forall x \in F,\ N(B(x, R) \cap F, r) \leq C \left( \frac{R}{r} \right)^s \right\}.$$

The quasi-Assouad dimension is obtained by imposing a restricted relation between the scales: $$\dim_{qA} F = \lim_{\delta \to 0} \inf \left\{ s : \exists C>0,\ \forall 0 < r < R < 1,\ R \leq r^\delta,\ \forall x \in F,\ N(B(x, R) \cap F, r) \leq C \left( \frac{R}{r} \right)^s \right\}.$$ As presented in the literature, this is equivalently written as

$$\dim_{qA} F = \lim_{\delta \to 0} \inf \left\{ s : \exists C>0,\ \forall 0 < r < R^{1/\delta} < R < 1,\ \forall x \in F,\ N(B(x, R) \cap F, r) \leq C \left(\frac{R}{r}\right)^s \right\}.$$

The defining feature is the permitted gap $r < R^{1/\delta}$ as $\delta \to 0$. In the language used in the literature, this allows a sub-exponential gap between $r$ and $N(E,r)$0, whereas the full Assouad dimension quantifies over all $N(E,r)$1. A common interpretation is therefore that quasi-Assouad dimension ignores certain “sub-exponential effects” that can force the full Assouad dimension upward (Fraser et al., 2018).

Equivalent parametrizations also appear. For example, one may define auxiliary functions $N(E,r)$2 through covering estimates with $N(E,r)$3 and then set $N(E,r)$4; this formulation was used in the original treatment of quasi-uniform disconnectedness and Moran sets (Lü et al., 2014).

2. Spectral interpretation

A central development is the connection with the Assouad spectrum. For $N(E,r)$5, Fraser and Yu’s Assouad spectrum is

$N(E,r)$6

The related upper Assouad spectrum is

$N(E,r)$7

The decisive structural statement is

$N(E,r)$8

and consequently

$N(E,r)$9

Thus the Assouad spectrum interpolates from the upper box-counting dimension at the left-hand side of its domain to the quasi-Assouad dimension at the right-hand side, not necessarily to the full Assouad dimension (Fraser et al., 2018).

This right-endpoint behavior corrects an initially natural but false expectation: although the spectrum was designed as an interpolation between box-counting and Assouad dimensions, its limiting value as $r$0 is in general the quasi-Assouad dimension. The same work also shows that the spectrum can display unexpectedly complicated regularity: it can be strictly concave, can exhibit phase transitions of any order, need not be piecewise differentiable, and need not be constant near $r$1. In particular, the spectrum may still vary arbitrarily close to its quasi-Assouad limit (Fraser et al., 2018).

The same spectral perspective extends to lower Assouad-type dimensions. For uniformly perfect sets in doubling metric spaces,

$r$2

which gives an equivalent and more accessible definition of the quasi-lower Assouad dimension and sharpens the analogy between the upper and lower theories (Chen et al., 2018).

3. Position among other dimensions and structural properties

For compact sets, quasi-Assouad dimension sits in the usual Assouad-type hierarchy: $r$3 In projection problems one also has the useful estimate

$r$4

These inequalities may all be strict (García et al., 2017, Feng et al., 9 Jan 2026).

The quasi-Assouad dimension is monotone under inclusion, finitely stable under unions, and invariant under bi-Lipschitz and quasi-Lipschitz mappings. In the formulation of Lü and Xi, $r$5, and the condition $r$6 implies quasi uniform disconnectedness, paralleling the classical fact that $r$7 implies uniform disconnectedness (Lü et al., 2014).

The relationship with tangents is subtler than in the classical Assouad setting. García and Hare showed that arbitrary weak tangents do not in general yield quasi-Assouad lower bounds, but suitably controlled generalized fast tangents do: if $r$8 is a nontrivial fast tangent of $r$9, then $E$0, and under additional hypotheses one also obtains $E$1. The same paper established that quasi-Assouad dimension may increase under Lipschitz mappings: there exists $E$2 with

$E$3

This sharply separates quasi-Assouad behavior from Hausdorff and box dimensions, which do not increase under Lipschitz maps (García et al., 2017).

The role of quasi-Assouad dimension in orthogonal projections is nevertheless strong. If $E$4 has $E$5, then the box and packing dimensions of $E$6 are preserved under orthogonal projections onto almost all $E$7-dimensional subspaces, and the threshold $E$8 is sharp (Falconer et al., 2019).

4. Deterministic classes and explicit formulae

For several deterministic fractal classes, the quasi-Assouad dimension is explicitly computable and often reveals how local inhomogeneity interacts with the underlying symbolic or product structure. In planar self-affine geometry, García and Hare showed that for extended Lalley–Gatzouras and Barański carpets the quasi-Assouad and Assouad dimensions coincide: $E$9 with the corresponding lower statement $$\dim_A F = \inf \left\{ s : \exists C>0 \text{ s.t. } \forall 0<r<R<1,\ \forall x \in F,\ N(B(x, R) \cap F, r) \leq C \left( \frac{R}{r} \right)^s \right\}.$$0. The same work also proved a dichotomy for subsets of $$\dim_A F = \inf \left\{ s : \exists C>0 \text{ s.t. } \forall 0<r<R<1,\ \forall x \in F,\ N(B(x, R) \cap F, r) \leq C \left( \frac{R}{r} \right)^s \right\}.$$1 with decreasing gaps: if $$\dim_A F = \inf \left\{ s : \exists C>0 \text{ s.t. } \forall 0<r<R<1,\ \forall x \in F,\ N(B(x, R) \cap F, r) \leq C \left( \frac{R}{r} \right)^s \right\}.$$2, then $$\dim_A F = \inf \left\{ s : \exists C>0 \text{ s.t. } \forall 0<r<R<1,\ \forall x \in F,\ N(B(x, R) \cap F, r) \leq C \left( \frac{R}{r} \right)^s \right\}.$$3, whereas if $$\dim_A F = \inf \left\{ s : \exists C>0 \text{ s.t. } \forall 0<r<R<1,\ \forall x \in F,\ N(B(x, R) \cap F, r) \leq C \left( \frac{R}{r} \right)^s \right\}.$$4, then $$\dim_A F = \inf \left\{ s : \exists C>0 \text{ s.t. } \forall 0<r<R<1,\ \forall x \in F,\ N(B(x, R) \cap F, r) \leq C \left( \frac{R}{r} \right)^s \right\}.$$5 (García et al., 2017).

A more general self-affine formula is available for dominated rectangular self-affine sets with arbitrary overlaps in the plane, provided the projection IFS satisfies the asymptotically weak separation condition. If $$\dim_A F = \inf \left\{ s : \exists C>0 \text{ s.t. } \forall 0<r<R<1,\ \forall x \in F,\ N(B(x, R) \cap F, r) \leq C \left( \frac{R}{r} \right)^s \right\}.$$6 is the attractor, $$\dim_A F = \inf \left\{ s : \exists C>0 \text{ s.t. } \forall 0<r<R<1,\ \forall x \in F,\ N(B(x, R) \cap F, r) \leq C \left( \frac{R}{r} \right)^s \right\}.$$7 its projection to the principal axis, and $$\dim_A F = \inf \left\{ s : \exists C>0 \text{ s.t. } \forall 0<r<R<1,\ \forall x \in F,\ N(B(x, R) \cap F, r) \leq C \left( \frac{R}{r} \right)^s \right\}.$$8 the symbolic fibres, then

$$\dim_A F = \inf \left\{ s : \exists C>0 \text{ s.t. } \forall 0<r<R<1,\ \forall x \in F,\ N(B(x, R) \cap F, r) \leq C \left( \frac{R}{r} \right)^s \right\}.$$9

These results are new even under the strong separation condition and yield explicit examples with

$$\dim_{qA} F = \lim_{\delta \to 0} \inf \left\{ s : \exists C>0,\ \forall 0 < r < R < 1,\ R \leq r^\delta,\ \forall x \in F,\ N(B(x, R) \cap F, r) \leq C \left( \frac{R}{r} \right)^s \right\}.$$0

Thus the gap between quasi-Assouad and Assouad dimensions is not merely an artifact of overlaps (Fraser et al., 2022).

For Moran constructions, the theory has progressed from explicit limsup-max formulae in the homogeneous setting to exact formulae under broader regularity hypotheses. In the recent formulation, if $$\dim_{qA} F = \lim_{\delta \to 0} \inf \left\{ s : \exists C>0,\ \forall 0 < r < R < 1,\ R \leq r^\delta,\ \forall x \in F,\ N(B(x, R) \cap F, r) \leq C \left( \frac{R}{r} \right)^s \right\}.$$1, then for a Moran set $$\dim_{qA} F = \lim_{\delta \to 0} \inf \left\{ s : \exists C>0,\ \forall 0 < r < R < 1,\ R \leq r^\delta,\ \forall x \in F,\ N(B(x, R) \cap F, r) \leq C \left( \frac{R}{r} \right)^s \right\}.$$2,

$$\dim_{qA} F = \lim_{\delta \to 0} \inf \left\{ s : \exists C>0,\ \forall 0 < r < R < 1,\ R \leq r^\delta,\ \forall x \in F,\ N(B(x, R) \cap F, r) \leq C \left( \frac{R}{r} \right)^s \right\}.$$3

Without assuming $$\dim_{qA} F = \lim_{\delta \to 0} \inf \left\{ s : \exists C>0,\ \forall 0 < r < R < 1,\ R \leq r^\delta,\ \forall x \in F,\ N(B(x, R) \cap F, r) \leq C \left( \frac{R}{r} \right)^s \right\}.$$4, one always has $$\dim_{qA} F = \lim_{\delta \to 0} \inf \left\{ s : \exists C>0,\ \forall 0 < r < R < 1,\ R \leq r^\delta,\ \forall x \in F,\ N(B(x, R) \cap F, r) \leq C \left( \frac{R}{r} \right)^s \right\}.$$5; under the condition

$$\dim_{qA} F = \lim_{\delta \to 0} \inf \left\{ s : \exists C>0,\ \forall 0 < r < R < 1,\ R \leq r^\delta,\ \forall x \in F,\ N(B(x, R) \cap F, r) \leq C \left( \frac{R}{r} \right)^s \right\}.$$6

one obtains $$\dim_{qA} F = \lim_{\delta \to 0} \inf \left\{ s : \exists C>0,\ \forall 0 < r < R < 1,\ R \leq r^\delta,\ \forall x \in F,\ N(B(x, R) \cap F, r) \leq C \left( \frac{R}{r} \right)^s \right\}.$$7; and exact formulae follow for quasi-normal or normal Moran sets, including homogeneous Moran sets as a special case (Miao et al., 12 Nov 2025).

The quasi-Assouad dimension also appears as one endpoint of a full interval of intermediate Assouad-like dimensions. For suitable Cantor-type constructions, García, Hare, and Mendivil produced examples with

$$\dim_{qA} F = \lim_{\delta \to 0} \inf \left\{ s : \exists C>0,\ \forall 0 < r < R < 1,\ R \leq r^\delta,\ \forall x \in F,\ N(B(x, R) \cap F, r) \leq C \left( \frac{R}{r} \right)^s \right\}.$$8

and, more strongly, sets for which the collection of intermediate $$\dim_{qA} F = \lim_{\delta \to 0} \inf \left\{ s : \exists C>0,\ \forall 0 < r < R < 1,\ R \leq r^\delta,\ \forall x \in F,\ N(B(x, R) \cap F, r) \leq C \left( \frac{R}{r} \right)^s \right\}.$$9-dimensions fills a non-trivial interval whose endpoints are $$\dim_{qA} F = \lim_{\delta \to 0} \inf \left\{ s : \exists C>0,\ \forall 0 < r < R^{1/\delta} < R < 1,\ \forall x \in F,\ N(B(x, R) \cap F, r) \leq C \left(\frac{R}{r}\right)^s \right\}.$$0 and $$\dim_{qA} F = \lim_{\delta \to 0} \inf \left\{ s : \exists C>0,\ \forall 0 < r < R^{1/\delta} < R < 1,\ \forall x \in F,\ N(B(x, R) \cap F, r) \leq C \left(\frac{R}{r}\right)^s \right\}.$$1 (García et al., 2019).

5. Random models and typical behavior

Random constructions are a primary source of strict separation between quasi-Assouad and Assouad dimensions. For stochastically self-similar random recursive sets satisfying the uniform open set condition and standard boundedness assumptions, the quasi-Assouad dimension is almost surely equal to the almost sure Hausdorff dimension: $$\dim_{qA} F = \lim_{\delta \to 0} \inf \left\{ s : \exists C>0,\ \forall 0 < r < R^{1/\delta} < R < 1,\ \forall x \in F,\ N(B(x, R) \cap F, r) \leq C \left(\frac{R}{r}\right)^s \right\}.$$2 The same almost sure equality is indicated for random homogeneous and $$\dim_{qA} F = \lim_{\delta \to 0} \inf \left\{ s : \exists C>0,\ \forall 0 < r < R^{1/\delta} < R < 1,\ \forall x \in F,\ N(B(x, R) \cap F, r) \leq C \left(\frac{R}{r}\right)^s \right\}.$$3-variable models. In these settings the quasi-Assouad dimension tracks the “generic” fractal geometry, whereas the Assouad dimension can remain maximal (Troscheit, 2017).

In non-conformal random geometry, the behavior is richer. For random self-affine Bedford–McMullen carpets generated with probabilities $$\dim_{qA} F = \lim_{\delta \to 0} \inf \left\{ s : \exists C>0,\ \forall 0 < r < R^{1/\delta} < R < 1,\ \forall x \in F,\ N(B(x, R) \cap F, r) \leq C \left(\frac{R}{r}\right)^s \right\}.$$4, column counts $$\dim_{qA} F = \lim_{\delta \to 0} \inf \left\{ s : \exists C>0,\ \forall 0 < r < R^{1/\delta} < R < 1,\ \forall x \in F,\ N(B(x, R) \cap F, r) \leq C \left(\frac{R}{r}\right)^s \right\}.$$5, maximal column occupancies $$\dim_{qA} F = \lim_{\delta \to 0} \inf \left\{ s : \exists C>0,\ \forall 0 < r < R^{1/\delta} < R < 1,\ \forall x \in F,\ N(B(x, R) \cap F, r) \leq C \left(\frac{R}{r}\right)^s \right\}.$$6, and grid widths $$\dim_{qA} F = \lim_{\delta \to 0} \inf \left\{ s : \exists C>0,\ \forall 0 < r < R^{1/\delta} < R < 1,\ \forall x \in F,\ N(B(x, R) \cap F, r) \leq C \left(\frac{R}{r}\right)^s \right\}.$$7, the almost sure quasi-Assouad dimension is

$$\dim_{qA} F = \lim_{\delta \to 0} \inf \left\{ s : \exists C>0,\ \forall 0 < r < R^{1/\delta} < R < 1,\ \forall x \in F,\ N(B(x, R) \cap F, r) \leq C \left(\frac{R}{r}\right)^s \right\}.$$8

This quantity is typically distinct from both the upper box dimension and the almost sure Assouad dimension, the latter being determined by maxima over the deterministic patterns rather than the probability weights (Fraser et al., 2018).

The generalized Assouad spectrum makes the transition between quasi-Assouad and Assouad regimes explicit. For several random models, including Galton–Watson boundaries and one-variable random self-similar and self-affine constructions, the threshold function

$$\dim_{qA} F = \lim_{\delta \to 0} \inf \left\{ s : \exists C>0,\ \forall 0 < r < R^{1/\delta} < R < 1,\ \forall x \in F,\ N(B(x, R) \cap F, r) \leq C \left(\frac{R}{r}\right)^s \right\}.$$9

governs a phase transition: above the threshold the generalized spectrum agrees with the quasi-Assouad dimension, whereas below it the spectrum agrees with the Assouad dimension (Troscheit, 2019).

Comparable threshold phenomena occur for random complementary sets in $r < R^{1/\delta}$0. Under a natural random rearrangement model and a level comparable gap sequence $r < R^{1/\delta}$1, the quasi-Assouad dimension is almost surely the same as that of the associated Cantor set, while the full Assouad dimension is almost surely $r < R^{1/\delta}$2 and the lower dimension $r < R^{1/\delta}$3. Here the quasi-Assouad dimension reflects typical gap statistics, whereas the Assouad dimension is dominated by rare extreme local configurations (García et al., 2019).

Typicality in Banach spaces yields another form of separation. In little $r < R^{1/\delta}$4-Hölder spaces, a typical graph has quasi-Assouad dimension $r < R^{1/\delta}$5. By contrast, in the space with modulus of continuity $r < R^{1/\delta}$6, a typical graph has Assouad dimension $r < R^{1/\delta}$7 but quasi-Assouad dimension equal to $r < R^{1/\delta}$8. This provides a clean example where the full Assouad dimension detects sparse spikes that the quasi-Assouad dimension systematically discounts (Feng et al., 9 Jan 2026).

6. Measure-theoretic analogues and further extensions

There is a direct measure-theoretic analogue. For a Borel probability measure $r < R^{1/\delta}$9, the upper quasi-Assouad dimension is defined through the least exponent $\delta \to 0$0 such that

$\delta \to 0$1

uniformly for $\delta \to 0$2 and $\delta \to 0$3, followed by the limit $\delta \to 0$4. The lower quasi-Assouad dimension is defined by reversing the inequality (Hare et al., 2018).

For measures, the quasi-Assouad dimension is bounded above by the Assouad dimension and below by the quasi-Assouad dimension of the support; it also dominates the supremum of the upper local dimensions, and all of these inequalities can be strict. In the important class of self-similar measures on $\delta \to 0$5 with full interval support and the weak separation condition, finite quasi-Assouad dimension is equivalent to quasi-doubling. For generalized regular self-similar measures satisfying the weak separation condition,

$\delta \to 0$6

The same framework produces examples where $\delta \to 0$7 but $\delta \to 0$8 (Hare et al., 2018).

Random measures behave in the opposite direction. Under a natural independent simplex-splitting model on $\delta \to 0$9, almost every measure has infinite upper quasi-Assouad dimension and zero lower quasi-Assouad dimension. This shows that finite quasi-Assouad dimension is highly nongeneric in the ambient space of all Borel probability measures and typically requires strong regularity (Shen, 2019).

The quasi-Assouad dimension of a measure sits inside the broader $r$0-dimension formalism. Intermediate Assouad-like dimensions for measures recover the Assouad dimensions when $r$1, the $r$2-Assouad spectrum for constant $r$3, and the quasi-Assouad dimension in the limit $r$4. For self-similar measures satisfying the strong separation condition, all upper and lower $r$5-dimensions coincide with the extrema of the local dimensions (Hare et al., 2020).

The lower theory is similarly well developed. If $r$6 is quasi-doubling, then the upper and lower Assouad spectra of $r$7 converge as $r$8 to the upper and lower quasi-Assouad dimensions, respectively. Moreover, for self-similar measures of finite type, the quasi-lower Assouad dimension equals the infimum of the lower local dimensions, while coincidence of upper and lower Assouad dimensions does not imply $r$9-regularity (Hare et al., 2018).

In contemporary fractal geometry, the quasi-Assouad dimension thus occupies a precise intermediate position: it is strong enough to detect localized inhomogeneity, weak enough to ignore certain isolated scale anomalies, and sufficiently flexible to admit exact formulae and threshold theorems across self-affine, Moran, random, projectional, and measure-theoretic settings.