On the Marstrand projection theorem for the Assouad spectrum

Abstract: Marstrand's projection theorem states that the Hausdorff dimension of the orthogonal projection of a Borel set in the plane onto lines is constant almost surely. This property extends to other notions of dimension, such as box and packing dimensions, but does not hold for the Assouad dimension. In this paper, we show that Marstrand's projection theorem also fails for the quasi-Assouad dimension and the Assouad spectrum, which interpolates between the upper box and quasi-Assouad dimensions. Additionally, we establish an almost sure lower bound for the Assouad spectrum of the projections using capacity-theoretic dimension profiles, and an almost sure upper bound for projections of bounded planar sets via an incidence geometry-inspired tube-counting argument. As an application, for a parametrised family of homogeneous self-similar sets, we obtain an almost sure upper bound for the Assouad spectrum which beats the trivial upper bound coming from the upper box dimension.

• The paper demonstrates the failure of Marstrand's theorem for the Assouad spectrum and quasi-Assouad dimensions via explicit counterexamples.
• It establishes refined almost sure lower and upper bounds using capacity profiles and incidence geometry arguments.
• It applies these results to self-similar sets, confirming improved spectrum bounds and highlighting open challenges for bounded sets.

The Marstrand Projection Theorem for the Assouad Spectrum: Failure, Bounds, and Applications

Introduction and Context

This paper rigorously analyzes the behavior of the Assouad spectrum and quasi-Assouad dimension under orthogonal projections, with a focus on analogues of Marstrand's projection theorem. Traditionally, Marstrand’s theorem addresses the Hausdorff dimension, establishing constancy (almost surely) of the dimension of projections of planar Borel sets. The same property holds for box and packing dimensions, but for more refined notions—namely, the Assouad and quasi-Assouad dimensions and the Assouad spectrum—earlier work has shown that almost sure constancy fails.

The main contributions are: demonstrating the sharp failure of Marstrand’s theorem for both the quasi-Assouad dimension and the Assouad spectrum; establishing refined almost sure lower and upper bounds for these dimensions in the context of orthogonal projections; and presenting applications to self-similar sets and explicit constructions of sets exhibiting “wild” projection dimension behavior. The investigation extends and interconnects geometric measure theory, dimension profiles via capacity theory, and incidence geometry arguments.

Dimensions, Projections, and Spectra: Main Definitions

The Assouad dimension $\dim_{\mathrm{A}} X$ quantifies the extreme local scaling properties of a set, defined via covering numbers $N_r(A)$ in balls $B(x, R)$ with radii ratio $r \ll R$, capturing the most extreme local densities. The Assouad spectrum $\dim_{\mathrm{A}}^\theta X$ parametrizes the dimension between the upper box dimension $\overline{\dim}_{\mathrm{B}} X$ (at $\theta \to 0$) and the Assouad dimension (at $\theta \to 1$), interpolating using parameter $\theta \in (0,1)$. The quasi-Assouad dimension is the limit as $\theta \uparrow 1$ of the spectrum.

Orthogonal projections $N_r(A)$0 for $N_r(A)$1 (the $N_r(A)$2-dimensional Grassmannian) are the primary focus, especially the effect of projection on the spectrum and dimensions of $N_r(A)$3. The classical Marstrand theorem yields for the Hausdorff dimension $N_r(A)$4 that, for almost all $N_r(A)$5,

$N_r(A)$6

However, for the Assouad spectrum and quasi-Assouad dimension, these identities do not generally hold.

Principal Results: Non-Constancy and Sharp Bounds

Failure of Marstrand-Type Constancy

The authors provide explicit constructions (notably Theorem 1.6 and 1.8) showing that for the Assouad spectrum and quasi-Assouad dimension (and several formulations of generalized upper box dimension), the projections $N_r(A)$7 can take arbitrarily many distinct values on disjoint open subsets of $N_r(A)$8 for unbounded, uniformly discrete sets $N_r(A)$9. This demonstrates maximal failure of Marstrand-type theorems: the spectrum can be forced to simultaneously take prescribed values on open sets of directions.

Almost Sure Lower Bounds via Dimension Profiles

Despite the lack of constancy, almost sure lower bounds for the spectrum are shown to be expressible in terms of dimension profiles defined via capacity theory:

$B(x, R)$0

For bounded sets $B(x, R)$1 and almost all $B(x, R)$2, Theorem 2.2 gives:

$B(x, R)$3

and further explicit lower bounds in terms of the spectrum of $B(x, R)$4 (Corollary 2.3). In the planar case, for bounded sets and for almost all $B(x, R)$5,

$B(x, R)$6

Almost Sure Upper Bounds via Incidence Geometry

A novel upper bound is proved for the planar case using a tube-counting argument inspired by incidence geometry. For bounded $B(x, R)$7 and $B(x, R)$8, almost all $B(x, R)$9 satisfy:

$r \ll R$0

(see Theorem 2.8). Stronger statements are made quantifying the Hausdorff dimension of the exceptional set where the bound may fail (Theorem 2.9).

Applications to Self-Similar Sets

The results are specialized to parametrized families of self-similar sets, where projections along varying directions yield 1-parameter families of sets $r \ll R$1. Assuming the planar “lift” has quasi-Assouad dimension matching its similarity dimension, it is shown (Theorem 2.13) that for almost all parameters $r \ll R$2 the Assouad spectrum of $r \ll R$3 admits a strictly improved upper bound over the trivial box dimension bound, for all $r \ll R$4 and for almost every parameter value. This directly supports conjectures (cf. [jon:book]) that self-similar sets under generic conditions have spectrum matching their Hausdorff dimension.

Explicit “Wild” Counterexamples

For unbounded, uniformly discrete sets in $r \ll R$5, the authors construct $r \ll R$6 such that the projected Assouad spectrum (or quasi-Assouad/upper box dimension, including generalized formulations) can take any prescribed finite or countable sequence of values, on open sets of directions. This is realized by placing “bricks” with carefully controlled local and global geometry and orientation, interleaving scales and directionality.

A technical analysis demonstrates that, for such sets, no Marstrand-type identity holds for the spectrum: the spectrum of projections is not only non-constant almost surely, but can be assigned any countable collection of values on measurable sets of directions with positive measure.

Implications and Theoretical Developments

Theoretical Significance

• Refinement of Dimension Theory: The work fully clarifies how Marstrand projection results break down for quasi-Assouad and Assouad spectrum, emphasizing the necessity of new profile-based invariants for sharp almost sure bounds.
• Spectrum Interpolation and Profiles: The role of dimension profiles as precise lower bounds prompts further investigation of their attainability, sharpness, and possible counterparts to the established profiles for box/packing dimension.
• Distinction Between Bounded and Unbounded Sets: The upper bounds for projections rely critically on global boundedness, indicating fundamentally different phenomena in dimension theory for unbounded sets in comparison to the classical (Hausdorff, box) situation.
• Incidence Geometry in Fractals: The use of tube-overlap and selection arguments from incidence geometry demonstrates the broader applicability of geometric-combinatorial reasoning in dimension theory.

Practical Implications and Open Directions

• Projection Behavior in Analysis and Dynamics: The results are directly relevant for understanding the almost sure dimension of images of sets or measures under random projections, with applications to embedding theory, fractal percolation, and signal analysis.
• Sharpness and Classification of Exceptional Sets: The precise Hausdorff dimension estimates (even zero-dimensionality or countability) of exceptional sets of directions suggest further refinements in “almost everywhere” statements for generalized dimensions.
• Spectrum for Structured and Random Fractals: The application to self-similar and potentially self-affine/random fractals opens avenues for understanding the full spectrum of typical projections in broader stochastic or deterministic settings.

Future Developments

• Marstrand-Type Identities for Bounded Sets: It remains open whether a Marstrand projection theorem for the Assouad spectrum might hold for bounded planar sets; there is no known example to the contrary, and sharper upper and lower profile inequalities could potentially yield an identity in this restricted regime.
• Capacity Profiles and Algorithmic Computation: The translation of spectrum characterizations into capacity-theoretic profiles, based on kernels and energy minimization, suggests feasibility for algorithmic computation or estimation of spectrum for compact sets—of considerable interest in applications.
• Interplay with Quasiconformal and Metric Geometry: The spectrum is known to have deep connections with quasiconformal mapping properties and weak embeddability; this work scaffolds further explorations into metric invariance, distortion, and rigidity phenomena.

Conclusion

This work provides a comprehensive and technically sharp analysis of projection theorems for the Assouad spectrum, demonstrating both the limits of Marstrand-type results and the structure of almost sure bounds in terms of dimension profiles and incidence geometry. The counterexamples for unbounded sets solidify the essential distinctions among various dimension notions, while the profile bounds furnish a precise toolkit for bounding dimensions of typical projections, with important applications to fractal geometry, harmonic analysis, and dynamical systems. The question of a Marstrand theorem for bounded sets remains open and stands as a significant challenge for future research.

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Reference: "On the Marstrand projection theorem for the Assouad spectrum" (2606.28830)