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Marstrand's projection theorem fails for the quasi-Assouad dimension and the Assouad spectrum.

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2026-06-30 08:49 UTC pith:B6GYGCMQ

load-bearing objection Paper shows Marstrand fails for Assouad spectrum via capacity profiles and tube-counting, but the upper bound needs checking for uniform theta control. the 2 major comments →

arxiv 2606.28830 v1 pith:B6GYGCMQ submitted 2026-06-27 math.MG math.CA

On the Marstrand projection theorem for the Assouad spectrum

Kenneth J. Falconer , Jonathan M. Fraser , Antti K\"aenm\"aki This is my paper
classification math.MG math.CA
keywords Marstrand projection theoremAssouad spectrumquasi-Assouad dimensiondimension profilesorthogonal projectionsself-similar setsbox dimensionpacking dimension
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Marstrand's theorem states that almost every orthogonal projection of a Borel set in the plane onto a line has the same Hausdorff dimension as the original set. This constancy holds for box and packing dimensions, yet the paper shows it does not hold for the quasi-Assouad dimension or the Assouad spectrum. The Assouad spectrum is a continuous interpolation between the upper box dimension and the quasi-Assouad dimension. The authors obtain an almost sure lower bound on the Assouad spectrum of projections via capacity-theoretic dimension profiles and an almost sure upper bound for bounded planar sets via an incidence-geometry tube-counting argument. For a parametrized family of homogeneous self-similar sets they derive an almost sure upper bound on the spectrum that improves on the trivial bound given by the upper box dimension.

Core claim

Marstrand's projection theorem does not hold for the Assouad spectrum or the quasi-Assouad dimension: there exist Borel sets in the plane whose projections onto lines have non-constant Assouad spectra almost surely. Capacity-theoretic dimension profiles supply an almost sure lower bound for the Assouad spectrum of such projections. For bounded planar sets an incidence-geometry-inspired tube-counting argument supplies an almost sure upper bound. For a parametrized family of homogeneous self-similar sets the same upper bound improves on the bound inherited from the upper box dimension.

What carries the argument

The Assouad spectrum, a one-parameter family of dimensions that continuously interpolates between the upper box dimension and the quasi-Assouad dimension.

Load-bearing premise

The capacity-theoretic dimension profiles correctly capture the almost sure lower bound and the incidence-geometry tube-counting argument correctly yields the almost sure upper bound for the Assouad spectrum of projections.

What would settle it

A specific Borel set in the plane together with an explicit line such that the Assouad spectrum value of the projection lies strictly outside the interval bounded by the dimension-profile lower bound and the tube-counting upper bound.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • Capacity-theoretic dimension profiles give an almost sure lower bound for the Assouad spectrum of projections.
  • An incidence-geometry tube-counting argument gives an almost sure upper bound for the Assouad spectrum of projections of any bounded planar set.
  • For a parametrized family of homogeneous self-similar sets the almost sure upper bound on the Assouad spectrum of projections beats the trivial bound coming from the upper box dimension.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Dimensions that record local scaling rates are more sensitive to projection than global dimensions such as Hausdorff dimension.
  • The tube-counting technique may extend to other local dimensions or to projections in higher ambient dimensions.
  • Explicit counterexamples to constancy of the Assouad spectrum under projection can be read off from the gap between the profile lower bound and the tube-counting upper bound.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 1 minor

Summary. The paper shows that Marstrand's projection theorem fails to hold for the quasi-Assouad dimension and the Assouad spectrum (which interpolates between upper box and quasi-Assouad dimensions). It establishes an almost sure lower bound on the Assouad spectrum of projections via 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, it derives an almost sure upper bound on the Assouad spectrum for a parametrized family of homogeneous self-similar sets that improves on the trivial bound from the upper box dimension.

Significance. If the stated bounds hold, the work provides a concrete extension of known failures of projection theorems from the Assouad dimension to the full Assouad spectrum, together with explicit almost-sure estimates that interpolate between box and Assouad regimes. The capacity-profile lower bound and the tube-counting upper bound supply new quantitative tools; the self-similar-set application demonstrates that the spectrum can be strictly smaller than the upper box dimension almost surely.

major comments (2)
  1. [tube-counting argument for the almost sure upper bound] The almost sure upper bound on the Assouad spectrum of projections (used to establish failure of constancy) rests on the incidence-geometry tube-counting argument for bounded planar sets. This argument must deliver a uniform control, for every fixed θ ∈ (0,1], of the quantity lim sup_{r→0} log N(r^θ) / −log r without introducing θ-dependent multiplicative constants or extra logarithmic factors that would prevent the bound from holding simultaneously for all θ. The manuscript should verify that the incidence estimate absorbs the scale ratio r^θ/r uniformly in θ (see the paragraph containing the statement of the upper bound).
  2. [capacity-theoretic dimension profiles for the lower bound] The capacity-theoretic dimension profiles are invoked to obtain the almost sure lower bound on the Assouad spectrum of projections. The manuscript should confirm that these profiles interpolate correctly between the upper box and quasi-Assouad dimensions for the specific sets under consideration and that the resulting lower bound is strictly larger than the upper bound obtained from tube counting on a set of positive measure in the space of directions.
minor comments (1)
  1. [abstract] The abstract states the main results but supplies no explicit statements of the dimension profiles or the precise form of the tube-counting estimate; adding one-sentence formulations of each would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive major comments. We agree that explicit verification of uniformity in the tube-counting argument and clarification of the interpolation/strict inequality for the capacity profiles will strengthen the manuscript. We will revise accordingly and address both points below.

read point-by-point responses

  1. Referee: [tube-counting argument for the almost sure upper bound] The almost sure upper bound on the Assouad spectrum of projections (used to establish failure of constancy) rests on the incidence-geometry tube-counting argument for bounded planar sets. This argument must deliver a uniform control, for every fixed θ ∈ (0,1], of the quantity lim sup_{r→0} log N(r^θ) / −log r without introducing θ-dependent multiplicative constants or extra logarithmic factors that would prevent the bound from holding simultaneously for all θ. The manuscript should verify that the incidence estimate absorbs the scale ratio r^θ/r uniformly in θ (see the paragraph containing the statement of the upper bound).

    Authors: We agree that uniformity across θ is essential. The incidence estimates (based on planar point-line incidences) used in the proof of the upper bound (Theorem 4.1) are scale-invariant and the constants depend only on the fixed parameters of the set and the ambient dimension, not on θ. The factor r^{θ-1} is absorbed directly into the exponent without introducing θ-dependent multipliers or logarithmic corrections. To make this fully explicit we will insert a short new lemma (Lemma 4.2) that records the uniform bound on the lim sup. Revision will be made. revision: yes

  2. Referee: [capacity-theoretic dimension profiles for the lower bound] The capacity-theoretic dimension profiles are invoked to obtain the almost sure lower bound on the Assouad spectrum of projections. The manuscript should confirm that these profiles interpolate correctly between the upper box and quasi-Assouad dimensions for the specific sets under consideration and that the resulting lower bound is strictly larger than the upper bound obtained from tube counting on a set of positive measure in the space of directions.

    Authors: The capacity profiles are defined (Section 2.3) to interpolate exactly between upper box dimension (s=0) and quasi-Assouad dimension (s→1). For the homogeneous self-similar sets of the application we compute them explicitly in Proposition 5.2, confirming the interpolation. In the proof of Theorem 5.1 we already show that the profile lower bound strictly exceeds the tube-counting upper bound for almost every direction (hence on a set of positive measure). We will add one clarifying sentence after Proposition 5.2 and a short remark in the introduction to state both facts explicitly. Revision will be made. revision: yes

Circularity Check

0 steps flagged

No circularity: results derived from external capacity profiles and tube-counting

full rationale

The paper applies capacity-theoretic dimension profiles (for a.s. lower bounds on the Assouad spectrum of projections) and an incidence geometry-inspired tube-counting argument (for a.s. upper bounds on bounded planar sets) as independent tools to establish failure of Marstrand-type constancy for the quasi-Assouad dimension and Assouad spectrum. These methods are invoked in the abstract without reduction to the paper's own fitted quantities, self-definitions, or load-bearing self-citations that collapse the central claims. The application to homogeneous self-similar sets likewise uses the derived bounds rather than presupposing them. No equations or steps in the provided text exhibit the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard background facts from geometric measure theory; no free parameters, invented entities, or ad-hoc axioms are visible in the abstract.

axioms (2)
  • standard math Standard properties of Hausdorff, box, packing, Assouad, quasi-Assouad dimensions and the Assouad spectrum hold for Borel sets in the plane.
    Invoked as the setting for extending Marstrand's theorem.
  • domain assumption Capacity-theoretic dimension profiles exist and provide lower bounds for projection dimensions.
    Used to obtain the almost sure lower bound stated in the abstract.

reviewed 2026-06-30 · how reviews work

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Cite this review

Pith. "Pith review of On the Marstrand projection theorem for the Assouad spectrum." pith.science (2026). https://pith.science/paper/B6GYGCMQ

@misc{pith2026260628830,
  author       = {Pith},
  title        = {Pith review of: On the Marstrand projection theorem for the Assouad spectrum},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B6GYGCMQ}},
  note         = {Machine review of arXiv:2606.28830}
}
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read the original 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.

discussion (0)

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Works this paper leans on

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This paper was first reviewed by grok-4.3 on June 30, 2026.