Generalized Upper Box Dimension

• Generalized upper box dimension is defined as the small-θ limit of the Assouad spectrum, extending the classical upper box dimension to arbitrary subsets including unbounded sets.
• It bridges local covering strategies with global geometric insights, linking properties of packing and Assouad dimensions through precise spectral bounds.
• It preserves key structural properties such as monotonicity, stability under finite unions, bi-Lipschitz invariance, and consistency with classical theory on bounded sets.

Searching arXiv for the core paper and closely related work on generalized upper box dimension and neighboring upper-box generalizations. {"query":"ti:\"The generalized upper box dimension\" OR (Wang et al., 1 Oct 2025)", "max_results": 10} Generalized upper box dimension is a dimension notion for arbitrary subsets of $\mathbb{R}^d$, including unbounded sets, defined by the small-$\theta$ limit of the Assouad spectrum. It was introduced to remove the boundedness restriction built into the classical upper box dimension, while preserving the latter on bounded sets and retaining natural compatibility with packing and Assouad-type dimensions. In its defining form,

$$\overline{\dim}_{GB} F=\lim_{\theta\to 0}\dim_A^\theta F=\lim_{\theta\to 0}\overline{\dim}_A^\theta F,$$

and for bounded $F\subset \mathbb{R}^d$ it coincides with the usual upper box dimension $\overline{\dim}_B F$ (Wang et al., 1 Oct 2025).

1. Classical background and the need for extension

For a nonempty bounded set $E\subset \mathbb{R}^d$, the classical upper box dimension is

$$\overline{\dim}_{B} E=\limsup_{\delta\to 0}\frac{\log N_\delta(E)}{-\log\delta},$$

where $N_\delta(E)$ is the smallest number of closed balls of radius $\delta$ required to cover $E$. This definition is naturally restricted to bounded sets: if $\theta$0 is unbounded, then $\theta$1 for every sufficiently small $\theta$2, so the formula ceases to be informative (Wang et al., 1 Oct 2025).

This boundedness obstruction contrasts with Hausdorff, packing, and Assouad dimensions, which are defined for arbitrary sets. The generalized upper box dimension addresses that mismatch by replacing direct global covering counts with a spectral definition coming from local covering behavior at separated scales (Wang et al., 1 Oct 2025).

A more naive unbounded extension considered in the literature is

$\theta$3

This quantity can be strictly smaller than $\theta$4. An explicit example is

$\theta$5

for which

$\theta$6

This shows that truncation by large Euclidean balls can miss scale-sensitive complexity distributed at infinity (Wang et al., 1 Oct 2025).

2. Definition through Assouad and upper spectra

The Assouad dimension of $\theta$7 is

$\theta$8

Its spectral refinements are the Assouad spectrum

$\theta$9

and the upper spectrum

$$\overline{\dim}_{GB} F=\lim_{\theta\to 0}\dim_A^\theta F=\lim_{\theta\to 0}\overline{\dim}_A^\theta F,$$0

The quasi-Assouad dimension is

$$\overline{\dim}_{GB} F=\lim_{\theta\to 0}\dim_A^\theta F=\lim_{\theta\to 0}\overline{\dim}_A^\theta F,$$1

(Wang et al., 1 Oct 2025).

A preliminary result shows

$$\overline{\dim}_{GB} F=\lim_{\theta\to 0}\dim_A^\theta F=\lim_{\theta\to 0}\overline{\dim}_A^\theta F,$$2

which motivates the definition

$$\overline{\dim}_{GB} F=\lim_{\theta\to 0}\dim_A^\theta F=\lim_{\theta\to 0}\overline{\dim}_A^\theta F,$$3

The limit of $$\overline{\dim}_{GB} F=\lim_{\theta\to 0}\dim_A^\theta F=\lim_{\theta\to 0}\overline{\dim}_A^\theta F,$$4 as $$\overline{\dim}_{GB} F=\lim_{\theta\to 0}\dim_A^\theta F=\lim_{\theta\to 0}\overline{\dim}_A^\theta F,$$5 is later shown to exist, so the defining limsup may be replaced by an actual limit (Wang et al., 1 Oct 2025).

The key interpolation inequalities are

$$\overline{\dim}_{GB} F=\lim_{\theta\to 0}\dim_A^\theta F=\lim_{\theta\to 0}\overline{\dim}_A^\theta F,$$6

and

$$\overline{\dim}_{GB} F=\lim_{\theta\to 0}\dim_A^\theta F=\lim_{\theta\to 0}\overline{\dim}_A^\theta F,$$7

These inequalities place generalized upper box dimension at the lower endpoint of the Assouad and upper spectra, exactly as classical upper box dimension sits at the small-$$\overline{\dim}_{GB} F=\lim_{\theta\to 0}\dim_A^\theta F=\lim_{\theta\to 0}\overline{\dim}_A^\theta F,$$8 endpoint for bounded sets (Wang et al., 1 Oct 2025).

3. Core properties

The generalized upper box dimension inherits the standard structural properties expected of a geometric dimension. It is monotone: $$\overline{\dim}_{GB} F=\lim_{\theta\to 0}\dim_A^\theta F=\lim_{\theta\to 0}\overline{\dim}_A^\theta F,$$9 It is stable under finite unions: $F\subset \mathbb{R}^d$0 It is bi-Lipschitz invariant, and it is unchanged by Euclidean closure: $F\subset \mathbb{R}^d$1 It also satisfies the product inequality

$F\subset \mathbb{R}^d$2

(Wang et al., 1 Oct 2025).

Its location among familiar dimensions is given by

$F\subset \mathbb{R}^d$3

The upper bound follows through the quasi-Assouad dimension, since $F\subset \mathbb{R}^d$4. The lower bound connects generalized upper box dimension to packing-type decomposition theory (Wang et al., 1 Oct 2025).

Consistency with classical theory is exact on bounded sets: $F\subset \mathbb{R}^d$5 Thus the new definition is an extension rather than a replacement of the upper box dimension (Wang et al., 1 Oct 2025).

4. Modified generalized upper box dimension and packing dimension

The modified generalized upper box dimension is defined by

$F\subset \mathbb{R}^d$6

An equivalent formulation may restrict the covering pieces $F\subset \mathbb{R}^d$7 to be bounded. The equivalence is proved by subdividing $F\subset \mathbb{R}^d$8 into half-open unit cubes and replacing each $F\subset \mathbb{R}^d$9 by bounded pieces $\overline{\dim}_B F$0, using monotonicity to preserve the relevant supremum (Wang et al., 1 Oct 2025).

This modified version satisfies the exact identity

$\overline{\dim}_B F$1

That theorem mirrors the classical relation between packing dimension and modified upper box dimension, and it is one of the main reasons the generalized upper box dimension is described as a reasonable generalization of the upper box dimension (Wang et al., 1 Oct 2025).

The identity also clarifies the division of labor between the two notions. The unmodified generalized upper box dimension is a direct extension of upper box dimension to arbitrary sets; the modified version plays the same decomposition-stable role that modified upper box dimension plays in the bounded theory, and therefore recovers packing dimension (Wang et al., 1 Oct 2025).

5. Zero-dimensional and full-dimensional spectral phenomena

The relation to quasi-Assouad dimension becomes especially sharp at the zero threshold: $\overline{\dim}_B F$2 Although $\overline{\dim}_B F$3 and $\overline{\dim}_B F$4 need not agree numerically in general, their vanishing is equivalent (Wang et al., 1 Oct 2025).

The same paper derives extremal equivalences for the upper spectrum and Assouad spectrum. For any $\overline{\dim}_B F$5,

$\overline{\dim}_B F$6

and

$\overline{\dim}_B F$7

Thus the two spectra may differ in intermediate regimes, but they agree at the minimal and maximal possible values (Wang et al., 1 Oct 2025).

These statements show that generalized upper box dimension is not merely an auxiliary endpoint quantity. Through the spectral bounds, it controls both the zero-dimensional regime and the way full ambient dimension propagates between the exact-scale and upper-threshold spectra. This suggests a particularly close relationship between generalized upper box dimension and the boundary behavior of Assouad-type interpolation families.

6. Broader research landscape

A distinct line of work uses the phrase “generalized upper box dimension” in a broader interpretive sense rather than as a single fixed definition. The most developed example is the $\overline{\dim}_B F$8-Assouad theory, where $\overline{\dim}_B F$9-Assouad dimensions interpolate between upper box and Assouad dimensions by prescribing a scale gap $E\subset \mathbb{R}^d$0. For any bounded doubling metric space $E\subset \mathbb{R}^d$1 and any

$E\subset \mathbb{R}^d$2

there exists a dimension function $E\subset \mathbb{R}^d$3 such that

$E\subset \mathbb{R}^d$4

and the authors explicitly interpret the $E\subset \mathbb{R}^d$5-family as a generalized upper box theory indexed by scale sensitivity (Banaji et al., 2023). In that framework, the upper variant is determined by the ordinary $E\subset \mathbb{R}^d$6-Assouad dimensions through

$E\subset \mathbb{R}^d$7

so it does not add an independent invariant (Banaji et al., 2023).

Another neighboring direction defines persistent-homology-based dimensions from extremal finite subsets. For bounded $E\subset \mathbb{R}^d$8 with

$E\subset \mathbb{R}^d$9

the first persistent-homology dimension satisfies

$$\overline{\dim}_{B} E=\limsup_{\delta\to 0}\frac{\log N_\delta(E)}{-\log\delta},$$0

while in degree $$\overline{\dim}_{B} E=\limsup_{\delta\to 0}\frac{\log N_\delta(E)}{-\log\delta},$$1 the persistent-homology dimension recovers upper box dimension for all metric spaces (Schweinhart, 2018). This is not the same notion as $$\overline{\dim}_{B} E=\limsup_{\delta\to 0}\frac{\log N_\delta(E)}{-\log\delta},$$2, but it is another explicit attempt to recast upper-box behavior in a more structural language.

A further neighboring literature does not redefine upper box dimension itself, but generalizes the classes of objects for which sharp upper-box formulas are available. For inhomogeneous self-similar sets, one has in general

$$\overline{\dim}_{B} E=\limsup_{\delta\to 0}\frac{\log N_\delta(E)}{-\log\delta},$$3

with equality under the strong open set condition (Fraser, 2013). In the presence of overlaps, the expected max formula can fail, and new upper bounds involving overlap-sensitive parameters become necessary (Baker et al., 2015). For inhomogeneous self-affine sets, the upper box dimension is bounded above by the maximum of the affinity dimension and the dimension of the condensation set (Burrell et al., 2018). These results do not define generalized upper box dimension in the strict sense of (Wang et al., 1 Oct 2025), but they illustrate how upper-box behavior naturally acquires generalized forms once boundedness, overlap, anisotropy, or nonuniform scale structure are introduced.

Taken together, these developments indicate two complementary meanings of the term. In the strict sense, generalized upper box dimension is the Assouad-spectrum-based extension

$$\overline{\dim}_{B} E=\limsup_{\delta\to 0}\frac{\log N_\delta(E)}{-\log\delta},$$4

for arbitrary sets (Wang et al., 1 Oct 2025). In a broader sense, the phrase also names a research program in which upper box dimension is extended, interpolated, or structurally reinterpreted to capture scale-sensitive geometric complexity beyond the classical bounded-set setting (Banaji et al., 2023).