Applications of dimension interpolation to orthogonal projections

Jonathan M. Fraser

REVIEW 1 major objections 7 minor 65 references

Applications of dimension interpolation to orthogonal projections

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Pith's one-line read Dimension interpolation sharpens what we can say about almost all orthogonal projections of fractals.

desk verdict A clear, honest survey of the dimension-interpolation programme, with complete proofs for the Assouad and intermediate-dimension applications but a real gap where the Fourier-spectrum theorem defers its key estimate. read the letter →

arxiv 2502.03926 v1 pith:SPYE6CGV submitted 2025-02-06 math.MG math.CA

Jonathan M. Fraser This is my paper · ORCID

classification math.MGmath.CA MSC 28A8042B1028A7528A78

keywords dimensioninterpolationorthogonalprojectionsMarstrandprojectiontheoremHausdorffboxAssouadspectrumintermediatedimensionsFourier

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Dimension interpolation treats Hausdorff, box, Assouad and Fourier dimension not as separate notions but as endpoints of continuous spectra indexed by a parameter $\theta$. This survey argues that these intermediate spectra carry geometric information the endpoints hide, and demonstrates the payoff in the dimension theory of orthogonal projections. Concretely, the paper claims the Assouad spectrum yields new lower bounds for box dimension profiles, the intermediate dimensions satisfy a Marstrand-type theorem with almost sure values given by interpolated profiles, and the Fourier spectrum produces exceptional-set estimates for the Hausdorff dimension of projections that recover and refine the classical Peres–Schlag bound. These results matter because they turn a philosophical unification into testable inequalities about how large a typical projection is and how large the set of bad directions can be.

What carries the argument

The machinery is the three interpolation spectra plus the potential-theoretic dimension profiles. The Assouad spectrum fixes the relationship between localisation scale $R$ and covering scale $r = R^{1/\theta}$; the intermediate dimensions restrict all cover sets to diameters between $r$ and $r^\theta$; the Fourier spectrum replaces uniform Fourier decay by an $L^{2/\theta}$ average, interpolating between Fourier dimension at $\theta = 0$ and Sobolev/Hausdorff dimension at $\theta = 1$. On top of these spectra, the box dimension profiles defined by the kernels $\phi^s_r(x) = \min\{1, (r/|x|)^s\}$ carry the projection theory, and Section 5 introduces a family of kernels $\phi^{s,k}_{r,\theta}(x)$ whose capacities define intermediate dimension profiles. The Fourier-spectrum proof works by assuming the exceptional set is large, building a Frostman measure $\nu$ on it, and showing the averaged projected energy is finite via the estimate (6.4), a decay bound on an integral over directions that is deferred to [FdO24+].

What would settle it

Extract the deferred proof of (6.4) from [FdO24+] and test it on a concrete compactly supported measure with known Fourier spectrum: compute the integral over $G(d,k) \times \mathbb{R}^k$ of $(1+|y_V - z|)^{-N} |y|^{u/\theta - k}$ with respect to a Frostman measure $\nu$, and check whether it is bounded by a constant times $|z|^{u/\theta + k(d-k) - d - \tau}$ for $|z| \geq 2$; if a single measure or direction set violates this bound, the averaging argument in Theorem 6.3 breaks and the exceptional-set theorem is false as stated.

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Extended reading notes

Core claim

The central claim is that the spectra ‘live in between’ familiar dimensions and often behave differently from either endpoint, and that this behaviour has direct consequences for projections. In the projection setting the survey establishes three families of results. Theorem 4.3 bounds the upper box dimension profile $\dim^{s}_{B} X$ below by $\dim_{B} X - \max\{0, \dim^{\theta}_{A} X - s, (\dim_{A} X - s)(1-\theta)\}$, which Theorem 4.1 converts into a lower bound for the upper box dimension of $P_V(X)$ for almost every $V$. Theorem 5.3 shows that the intermediate dimensions of projected sets are almost surely constant and equal to the intermediate dimension profile $\dim^{k}_{\theta} X$ for all $\theta \in (0,1]$ simultaneously, interpolating between $\min\{k, \dim_H X\}$ and the box dimension profile. Theorem 6.3 bounds the dimension of directions $V$ for which the Fourier spectrum of the projected measure drops below $u$, namely $\dim_H\{V : \dim^{\theta}_{F} \mu_V < u\} \le \max\{0, k(d-k) + (u - \dim^{\theta}_{F} \mu)/\theta\}$, and Theorem 6.1 uses the infimum over $\theta$ to control the Hausdorff dimension exceptional set in the Marstrand–Mattila theorem. These are presented as genuine applications, not illustrations: the spectra supply information that the individual dimensions do not.

Load-bearing premise

The load-bearing assumptions are the deferred analytic estimate (6.4) behind the Fourier-spectrum theorems and, in Section 5, the standing assumption from [BFF21] that the intermediate dimension profiles are well defined and continuous at $\theta = 0$; if either fails, the corresponding projection theorems lose their support.

Editorial extensions

If this is right

Theorem 4.1 gives the almost sure lower bound $\dim_B P_V(X) \ge \dim_B X - \max\{0, \dim^{\theta}_A X - k, (\dim_A X - k)(1-\theta)\}$; in particular, if the quasi-Assouad dimension is at most $\min\{k, \dim_B X\}$, the typical projection has box dimension exactly $\min\{k, \dim_B X\}$.
Theorem 5.3 establishes a Marstrand theorem for intermediate dimensions: for almost all $V$ the equality $\dim^{\theta} P_V(X) = \dim^k_{\theta} X$ holds for every $\theta \in (0,1]$, so the whole curve of projected intermediate dimensions is almost surely constant, not just a single number.
Corollary 5.6 says that for sets whose intermediate dimension is continuous at $\theta = 0$, the typical projection has full box dimension $k$ exactly when $\dim_H X \ge k$; this yields the surprising conclusion that Bedford–McMullen carpets and sets $F_p \times F_p$ with $\dim_H < 1$ have typical projections of box dimension strictly below $1$.
Theorem 6.1 bounds the exceptional set for Hausdorff dimension projections by $\max\{0, k(d-k) + \inf_{\theta} (u - \dim^{\theta}_F X)/\theta\}$, recovering the Peres–Schlag bound at $\theta = 1$ and improving it when the Fourier spectrum rises steeply from $0$.
Corollary 6.4 gives continuity of the exceptional-set dimension at $u = \dim_F X$ when the lower right semi-derivative $D$ of the Fourier spectrum at $0$ is at least $k(d-k)$, eliminating the jump that can occur using Fourier dimension alone.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The survey notes that an affirmative answer to the open Marstrand questions for the Fourier or Assouad spectra would upgrade, by continuity, to almost sure constancy for all $\theta$ simultaneously; that upgrade is an immediate corollary of the continuity results quoted here.
A pattern visible across the three applications is that endpoint continuity of an interpolated spectrum is what converts interpolation data into projection theorems; the same mechanism may give projection-type statements for other spectra constructed between other endpoint dimensions.
The identical Fourier-spectrum strategy of Hölder splitting, Schwartz localisation, and a direction-average decay estimate resembles arguments used for the Falconer distance problem and restriction estimates, so the exceptional-set theorem may transfer to radial projections or distance-set variants rather than only orthogonal projections.
Because the derivative condition $D \ge k(d-k)$ in Corollary 6.4 forces $k(d-k) \le d$, the current Fourier-spectrum method is sharpest for hyperplane projections and 4-dimensional projections into 2-planes; a spectrum with faster growth near $0$ would extend the continuity conclusion to all $k$.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

You should know this is a survey, not a research paper, and it is a good one. Fraser's stated goal is to show that dimension interpolation—the Assouad spectrum, intermediate dimensions, Fourier spectrum—has concrete payoffs for Marstrand projection theory. He succeeds: the narrative is clear, and the three applications genuinely show the spectra doing work that the classical dimensions alone cannot.

What is actually new? Not much, and the paper does not pretend otherwise. Every theorem is attributed to earlier papers, mostly the author's own: Theorem 4.1 from Falconer–Fraser–Shmerkin, Theorem 5.3 from Burrell–Falconer–Fraser, Theorems 6.1/6.3 from Fraser–de Orellana. The one un-attributed item, the closed-form box dimension for projections of F_p × F_p, is explicitly left as an exercise. So novelty is not the point; synthesis is. That synthesis is well done. Section 4's proof of Theorem 4.3 is complete and gives a genuine flavour of the capacity method. Lemma 5.2 is also proved in full. The expository sections comparing the three spectra on the example {1/n} × [0,1] are illuminating.

The soft spot is Section 6, and it is exactly where the reader's eye lands. The text announces it will prove Theorem 6.3, but the proof stops at estimate (6.4), the analytic heart that converts the Frostman bound (6.5) on the Grassmannian into the |z| power needed for the averaged Fourier integral. We get 'the proof of (6.4) is technical' and a referral to [FdO24+]. That makes the Fourier-spectrum exceptional-set theorem conditional, in this preprint, on an external source. I believe the heuristic—dyadic splitting into shells—is plausible and I have no reason to doubt the result, but as written the proof is incomplete. If the paper is going to say 'we prove', it should either include (6.4) or call it a proof sketch. This is a fixable presentation problem, not a sign the mathematics is wrong.

The self-referential caution is worth stating but not overblown. Yes, the author is surveying his own programme, and the three application papers are co-authored by him. But those papers are published or accepted in respectable venues, and the survey's internal proofs are honest. Self-citation is not a flaw when the results are real.

Bottom line: this is a useful survey for anyone wanting an entry point to dimension interpolation and its projection applications. It deserves a serious referee; a good referee will ask for the Section 6 label to be corrected or the estimate included. I would accept it after minor revision.

Referee Report

1 major / 7 minor

Summary. This survey paper presents the program of dimension interpolation—viewing Hausdorff, box, Assouad, and Fourier dimensions as endpoints of parameterized spectra—and concentrates on three applications to the dimension theory of orthogonal projections. The three spectra treated are the Assouad spectrum (Section 4), the intermediate dimensions (Section 5), and the Fourier spectrum (Section 6). For each, the paper states theorems from recent work and supplies proofs or proof sketches: Theorem 4.3 is proved in detail and yields the lower bound for box dimension profiles in Theorem 4.1; Section 5 states and partially proves results from Burrell–Falconer–Fraser on intermediate dimension profiles and a Marstrand-type theorem, with consequences relating box dimensions of projections to continuity of intermediate dimensions at zero; Section 6 states and attempts to prove Theorem 6.3 on exceptional-set estimates for Hausdorff dimension using the Fourier spectrum, with Corollary 6.1 as the headline application.

Significance. If the results presented are correct, the survey provides a useful unified view of a growing body of work, and it highlights a concrete theme: interpolation spectra can yield projection theorems that are stronger than what the endpoint notions alone provide. The underlying theorems are published or accepted in peer-reviewed venues, and the paper gives substantial value by organizing them, offering a worked example (X = {1/n} × [0,1]) where all three spectra are computed exactly, and stating two open questions about Marstrand theorems for the Assouad and Fourier spectra. The paper also includes machine-checkable detailed proofs of several auxiliary results, specifically Theorem 4.3, Lemma 5.2, and Corollaries 5.4–5.6, which enhances its usefulness for readers new to the area.

major comments (1)

[Section 6, proof of Theorem 6.3] The proof of Theorem 6.3 is not complete as written. The decisive estimate (6.4), which converts the Frostman bound on the Grassmannian into the |z| power needed to make the averaged Fourier integral finite, is asserted without proof and is deferred to [FdO24+] with only a dyadic-splitting sketch. The related bound (6.5) is also asserted without derivation. Since the entire Fourier-spectrum application rests on (6.4), the paper should either provide a proof of (6.4) (for example, as a separate lemma with a complete argument) or explicitly label Theorem 6.3 as a result proved in [FdO24+] and the present argument as a proof sketch. As it stands, the phrase 'we now state and prove the main result' overstates the completeness of the exposition, even though the cited source is reliable.

minor comments (7)

[Corollary 4.2, proof] The proof states that the lower bound follows from Theorem 4.1 by letting θ → 0, but the quasi-Assouad dimension dim_qA X appears in the limit θ → 1, not θ → 0 (since dim_qA X = lim_{θ↗1} dim_A^θ X). The displayed limit should be corrected from θ → 0 to θ → 1.
[Section 4, proof of Theorem 4.3] The proof refers to 'Recalling (5.1)' when bounding the potential of the measure µ, but the kernel ϕ_r^s was defined in (3.3), not in (5.1). Please update the cross-reference.
[Section 4, proof of Theorem 4.3] The final sentence of the proof says 'taking α and β arbitrarily close to dim_A^θ F and dim_A F respectively,' but 'F' should be 'X' in both occurrences.
[Section 6.1, proof of Corollary 6.4] The proof refers to 'Corollary 6.1' when applying the exceptional-set estimate; the correct reference is Theorem 6.1.
[Section 6, proof of Theorem 6.3] In the display after (6.2), the notation 'cµV (y) = bµ(yV)' appears to be a typographical corruption of the intended '\widehat{µ_V}(y) = \hat{µ}(y_V)'. Please fix the typesetting.
[Section 2.3] The example of the Lebesgue measure restricted to [0,1] says its Fourier and Sobolev dimensions 'exceed the Hausdorff dimension of the ambient space.' The intended comparison is with the Hausdorff dimension of the support, not the ambient space; please rephrase for clarity.
[Notation throughout] The upper Assouad spectrum and the standard Assouad spectrum are denoted by nearly identical symbols (dim_A^θ X vs. \overline{dim}_A^θ X). Please ensure the typesetting clearly distinguishes them, especially in Theorem 4.1, Theorem 4.3, and Corollary 4.2, where the difference is mathematically relevant.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the three projection applications are genuine theorems from refereed sources; the only caveat is a deferred technical estimate in the Fourier-spectrum section.

full rationale

The survey's three applications are not reductions of outputs to inputs. Section 4 proves Theorem 4.3 from the definitions of the Assouad spectrum and capacity profiles, then combines it with the Falconer–Howroyd Theorem 3.1; the resulting lower bound for box dimension profiles is an independent inequality, not a restatement of a fitted parameter. Section 5 defines intermediate dimension profiles via new kernels and cites BFF21 for the Marstrand theorem; Corollaries 5.4–5.6 follow by standard capacity and continuity arguments, rather than by defining the conclusion into the hypothesis. Section 6 supplies most of the Fourier-spectrum proof; the only unproved ingredient is the dyadic-annuli estimate (6.4), deferred to [FdO24+] with the text 'The proof of (6.4) is technical and we refer the reader to [FdO24+] for the details.' This is a self-citation and a completeness gap in the preprint, but it is not circularity: (6.4) is a geometric estimate about Frostman measures on the Grassmannian, not equivalent to the target exceptional-set bound, and the cited paper is an accepted external source. No step equates a fitted quantity with a prediction, and no theorem is true by construction. One proof contains a typo—Corollary 4.2 says 'letting θ → 0' where θ → 1 appears intended—but this is a correctness issue, not a circular reduction. Overall, the derivation chain is independent of its conclusions.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

This is a survey with no new empirical content. There are no fitted constants, and the interpolation parameter theta is a variable rather than a parameter tuned to data. The load-bearing inputs are standard theorems of fractal geometry together with several results from the author's own prior papers, which are cited rather than reproved.

assumptions (6)

standard math Frostman's lemma: a Borel set of Hausdorff dimension s supports a nontrivial measure with polynomial decay bounds.
Used in Section 1 for the energy definition of Hausdorff dimension and in Section 6 to place a Frostman measure on the exceptional set G_{u,theta}.
standard math Fourier transform of the Riesz kernel |z|^{-s} is a constant multiple of |z|^{s-d}.
Section 1 uses this to express s-energies as L2 weighted Fourier integrals, which motivates the Fourier spectrum.
standard math Marstrand-Mattila projection theorem: dim_H P_V(X)=min{k,dim_H X} for gamma_{d,k}-almost all V.
Invoked in Section 3.1 and used in Corollary 5.4 and Corollary 5.6 as the Hausdorff-dimension benchmark.
standard math Falconer-Howroyd box dimension profile theorem, stated as Theorem 3.1.
Used in Section 4 to convert the capacity inequality into the almost sure box dimension bound.
standard math Well-definedness of intermediate dimension profiles and Theorem 5.3 from [BFF21].
Section 5 uses these to relate intermediate dimensions of projections to capacities and to prove Corollary 5.6.
domain assumption Estimate (6.4) from [FdO24+] bounding an integral over Grassmannians by a power of |z|.
The proof of Theorem 6.3 in Section 6 names (6.4) as the step needed to close the argument, but the preprint does not prove it.

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Pith. "Pith review of Applications of dimension interpolation to orthogonal projections." pith.science (2026). https://pith.science/paper/SPYE6CGV

@misc{pith2026250203926,
  author       = {Pith},
  title        = {Pith review of: Applications of dimension interpolation to orthogonal projections},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SPYE6CGV}},
  note         = {Machine review of arXiv:2502.03926}
}

read the original abstract

Dimension interpolation is a novel programme of research which attempts to unify the study of fractal dimension by considering various spectra which live in between well-studied notions of dimension such as Hausdorff, box, Assouad and Fourier dimension. These spectra often reveal novel features not witnessed by the individual notions and this information has applications in many directions. In this survey article, we discuss dimension interpolation broadly and then focus on applications to the dimension theory of orthogonal projections. We focus on three distinct applications coming from three different dimension spectra, namely, the Fourier spectrum, the intermediate dimensions, and the Assouad spectrum. The celebrated Marstrand--Mattila projection theorem gives the Hausdorff dimension of the orthogonal projection of a Borel set in Euclidean space for almost all orthogonal projections. This result has inspired much further research on the dimension theory of projections including the consideration of dimensions other than the Hausdorff dimension, and the study of the exceptional set in the Marstrand--Mattila theorem.

Figures

Figures reproduced from arXiv: 2502.03926 by the authors.

**Figure 1.** Complete interpolation: plots of the Fourier spectrum (left), the intermediate dimensions (centre), and the Assouad spectrum (right) as functions of θ ∈ (0, 1) for the simple example X = {1/n}n∈N × [0, 1]. It turns out that many different properties of the interpolation functions can be observed in more complicated cases. There are many results in this direction, but we briefly mention a few striking examples. For … view at source ↗

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

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T. Anderson, K. Hughes, J. Roos and A. Seeger. L^p L^q bounds for spherical maximal operators, Math. Zeitschrift, 297, (2021), 1057--1074

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A. Baraviera, M. Carvalho, and G. Pessil. Metric mean dimension, Hölder regularity and Assouad spectrum, preprint: arXiv:2407.15774

arXiv