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Projections of self-affine fractals

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Pith's one-line read For Lebesgue almost every affine iterated function system with linearisation A, the dimension of each projected attractor equals a projected affinity dimension computed from the singular values, and the exceptional projections form…

desk verdict Strong paper extending Falconer's theorem to projections, with a load-bearing imported lemma on non-invertible projections that should be checked before acceptance. read the letter →

arxiv 2502.04001 v2 pith:RAX2PQFG submitted 2025-02-06 math.DS math.CAmath.MG

Ian Morris , Cagri Sert This is my paper · ORCID
classification math.DSmath.CAmath.MG MSC 28A8037C4537H15
keywords self-affinefractalsaffinitydimensionprojectiontheoremssingularvaluepressureexceptionalprojectionsequilibriumstatessumsetsexactdimensionality
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

This paper extends the 1988 theorem that for almost every self-affine set the affinity dimension gives the Hausdorff dimension, from the full set to every linear projection of it. The authors define a projected affinity dimension, $\dim^Q_{\mathrm{aff}} A$, computed from the singular values of the composites $QA_i$, and prove that under a pairwise contraction condition the Hausdorff dimension of $QX_v$ equals this number for Lebesgue almost every translation vector $v$, for each fixed projection $Q$. The projections split into finitely many algebraic families, each preserved by the group generated by the linear maps, and on each family the dimension is almost surely constant. The same machinery yields two further phenomena: equilibrium measures on self-affine sets whose projected measures are not exact-dimensional, and strongly irreducible self-affine sets with small sumsets but no arithmetic resonance.

What carries the argument

The carrying object is the projected singular-value potential and its pressure. For $s\in[0,\mathrm{rank}\,Q]$, the potential is $\varphi_s(QA)=\sigma_1(QA)\cdots\sigma_{\lfloor s\rfloor}(QA)\sigma_{\lceil s\rceil}(QA)^{s-\lfloor s\rfloor}$, and the projected pressure is $P_Q(A,s)=\lim_{n\to\infty}\frac1n\log\sum_{\lvert i\rvert=n}\varphi_s(QA_i)$. The projected affinity dimension $\dim^Q_{\mathrm{aff}} A$ is the unique zero of this strictly decreasing function. Theorem 1 shows that the pressure limit exists without subadditivity, satisfies a variational formula whose optimisers are $(\varphi_s,Q)$-equilibrium states, and has sublevel sets $\{Q:P_Q(A,s)\leq t\}$ that are algebraic varieties invariant under right multiplication by every $A_i$. The proof runs through an abstract theorem for potentials $\Psi_{U,b}$ built from operator norms on quotients of exterior-power representations of the Zariski-closed group generated by $A$, where quasi-multiplicativity yields unique $\psi$-mixing equilibrium states and a module-theoretic lemma makes the sublevel sets algebraic.

What would settle it

Compute the projected affinity dimension $\dim^Q_{\mathrm{aff}} A$ for a specific contracting tuple $A$ in $\mathbb{R}^4$ whose linearisation generates a Zariski-dense subsemigroup of $\mathbb{R}^*\mathrm{SO}(2,2)$, choose a rank-two isotropic projection $Q$, and measure the Hausdorff dimension of $QX_v$ for Lebesgue-random translations; a positive-measure set of $v$ with $\dim_H QX_v\neq\dim^Q_{\mathrm{aff}} A$ would falsify Theorem A(b), while equality for a single generic translation would support it.

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Extended reading notes

Core claim

The central claim is that Falconer's almost-sure dimension formula extends from attractors to their linear images. For every contracting tuple $A=(A_i)_{i\in I}$ and every $k=0,\ldots,d$, there exist integers $s_m<\cdots<s_1\leq k$ and a finite filtration $\varnothing=W_{m+1}\subset W_m\subset\cdots\subset W_1=\mathrm{Gr}(k,d)$ of algebraic varieties, each invariant under the Zariski closure of the semigroup generated by $A$, such that every affine IFS with linearisation $A$ has $\dim_H Q_U X\leq s_j$ whenever $U\in W_j\setminus W_{j+1}$. Under the additional hypothesis $\max_{i\neq j}\lVert A_i\rVert+\lVert A_j\rVert<1$, equality $\dim_H Q_U X=s_j$ holds for Lebesgue almost every affine IFS with that linearisation. The exponents $s_j$ are the zeros of a projected pressure function built from the singular values of $Q_UA_{i_1}\cdots A_{i_n}$, and the equality is proved for upper box dimension as well as for the local dimensions of projected equilibrium measures, giving a stratified projection theorem for generic self-affine sets.

Load-bearing premise

The almost-sure equalities in Theorem A(b) and Theorem 2(b) rest on the pairwise contraction assumption $\max_{i\neq j}\lVert A_i\rVert+\lVert A_j\rVert<1$, which the paper states cannot be removed entirely, and the proof also imports the integral estimate Lemma 5.2 from external sources rather than proving it.

Editorial extensions

If this is right

  • For every projection lying in a fixed algebraic level set of the projected affinity dimension, a Lebesgue-generic affine IFS with linearisation $A$ has $\dim_H QX_v$ equal to the corresponding exponent.
  • When the group generated by $A$ acts irreducibly on every exterior power $\wedge^\ell\mathbb{R}^d$, or when $A$ consists of similarities, the filtration is trivial, so for typical attractors every projection has the same dimension.
  • Theorem 2 implies a stratified projection theorem: on each algebraic variety $W_{t_j}$, for Lebesgue almost every translation vector the projected set has dimension $t_j$ for Lebesgue almost every projection in $W_{t_j}$.
  • The almost-sure statement cannot be upgraded to all projections, since the one-dimensional Sierpiński triangle has a countable dense family of exceptional projections from exact overlaps.
  • The method produces open sets of linearisations with nontrivial filtrations, giving a new general mechanism for constructing large families of exceptional projections of self-affine sets.

Reading between the lines

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

  • If Theorem A is correct, the exceptional projections of a typical self-affine set are organised by the representation theory of the Zariski closure of the semigroup generated by $A$, so classifying possible exceptional sets should reduce to decomposing exterior powers into simple $G^o$-modules.
  • Theorem B suggests that small sumsets of self-affine sets can arise from bundles of linear subspaces with nontrivial pairwise intersections rather than from arithmetic resonance; a testable extension would be to characterise the achievable dimension drop in terms of the intersection poset of those subbundles.
  • Theorem C implies that projected local Lyapunov dimensions need not be constant for ergodic measures, so any variational principle for dimensions of projected measures must treat essential suprema and essential infima separately rather than a single a.e.-constant value.
  • The finite bound $\prod_{k=1}^d(1+2\binom{d}{k})$ on the number of distinct level sets suggests that, for a fixed tuple $A$, the full list of exceptional-projection varieties is computable; testing small dimensions could reveal whether the bound is anywhere near sharp.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper extends Falconer's 1988 theorem on self-affine fractals to the dimensions of linear projections of the attractor. For a contracting linearization A, it introduces a Q-projected affinity dimension, proves a variational formula and algebraicity of its sublevel sets (Theorems 1 and 5), and shows that for Lebesgue almost every affine IFS with linearisation A, the projected attractor has Hausdorff and box dimension equal to the Q-projected affinity dimension (Theorem 2, Theorem A). The same machinery is used to construct new examples: equilibrium measures on self-affine fractals whose projections are not exact-dimensional (Theorem C) and strongly irreducible self-affine sets with small sumsets (Theorem B). The proofs rest on an abstract thermodynamic formalism for semigroup potentials and on a projected version of Falconer's integral estimate.

Significance. If the results hold, this is a major advance. Theorem A strictly generalizes Falconer's theorem and Marstrand's theorem in the self-affine setting, and the algebraic stratification of exceptional projections is a new structural phenomenon. Theorem C provides the first general examples of non-exact-dimensional projections of equilibrium measures on self-affine sets, and Theorem B gives small sumsets without arithmetic resonance. The abstract Theorem 5 is a substantial contribution in its own right. The paper is also commendably explicit about which ingredients are imported from previous work, which makes the remaining gaps visible and verifiable.

major comments (3)
  1. [§5.2, Lemma 5.2] The key integral estimate is stated with no proof: the text says it follows from the arguments of [8, §9.4] together with 'a simple extension' of [26, Lemma 2.2] to non-invertible Q. This lemma is load-bearing: it is the only place where the separation hypothesis max_{i≠j}||Ai||+||Aj||<1 enters Theorem 2(b), and the bound must hold with exactly C/φ_s(QA_{i∧j}), with no additional factor depending on the singular values of Q A_p. In the classical case Q=id the linear map from translation parameters to the difference is of full rank; for rank-deficient Q, the Jacobian degenerates on ker Q, so the stated extension is not a routine exercise. Since Theorem A(b), Corollary 2.3, and Theorem B all inherit this step, a complete proof or a precise published statement covering the non-invertible case is required.
  2. [§5.2, set-dimension lower bound] The proof asserts that from ess sup_x dimloc((QΠ^v)_*μ, x) ≥ dim^Q_aff A it follows that dim_H QX^v ≥ dim_* (QΠ^v)_*μ ≥ dim^Q_aff A. The second inequality is false in general: dim_* is the essential infimum of local dimension, not the essential supremum. Moreover, the chosen (φ_s,Q)-equilibrium state may have nonconstant pointwise Lyapunov dimension, as the paper itself emphasizes in the Technical Remark after Theorem 1. A correct proof must argue via positive-measure superlevel sets where the local dimension is at least dim^Q_aff A − ε and then apply the mass distribution principle; this repair seems feasible, but the current text does not supply it.
  3. [§5.2, measure clause of Theorem 2(b)] The lower-bound argument proves only dimloc ≥ sup{s : liminf (1/n log(φ_s(QA_{i|n})/μ([i|n]))) > 0}. The equality claimed in Theorem 2(b) with dim^Q_Lyap(A,μ)(i), defined as the infimum of s for which the pointwise limit is negative, is not established: if the pointwise limit equals zero on a nontrivial interval of s, the strict-positivity condition used to levy Lemma 5.3 fails, and the series ∑ μ([i|n])/φ_s(QA_{i|n}) need not converge. Since Theorem C is derived from this measure equality, the gap is load-bearing; the proof needs an argument covering the zero-limit plateau or a revised definition of the projected Lyapunov dimension.
minor comments (4)
  1. [§1.1] ‘F alconer’ and ‘Marstand’ are typographical errors for ‘Falconer’ and ‘Marstrand’; the same misspelling recurs in the header and elsewhere in the Introduction.
  2. [§3.4] In the proof of Theorem 5(b), the line ‘let F₂ ⊂ Γ_I be a finite set such that ⋃_{j∈F₁} G^o g_j = G’ appears to use the wrong index set (F₁ should be F₂); the subsequent definition F := F₂F₁ suggests a typo.
  3. [§5.2, Lemma 5.2] The statement of Lemma 5.2 does not specify the dependence of the constant C on r, Q, A, and the chosen norm; since the lemma is invoked with a fixed r, stating that C may depend on these parameters would avoid ambiguity.
  4. [Throughout] The abstract and Introduction repeatedly refer to ‘Marstand’s theorem’ and ‘Marstand problems’; standard spelling is ‘Marstrand’, which is also the spelling used in the bibliography.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the projected pressure and projected affinity dimension are defined independently of the conclusions, and the main lower bound rests on imported external estimates, not on the paper's own target claims.

full rationale

The central new object, P_Q(A,s), is defined directly as a limit of sums of φ_s(QA_i) in Section 2.2, and dim^Q_aff A is then defined as its zero (Definition 2.1). These definitions do not involve the Hausdorff dimension of QX or of any projected measure, so the equality statements in Theorem 2(b) and Theorem A are not true by construction. The lower-bound proof reduces, via Lemma 5.3, to the integral estimate Lemma 5.2, which is imported from [8, §9.4] and [26, Lemma 2.2]; these are independent prior results by other authors, not fitted to the paper's conclusions. The abstract machinery of Theorem 5 is proved in the paper using standard thermodynamic formalism (Theorem 4, cited to [21,31,68]) and a lemma from [56]; the latter has an indicated alternative proof via Hennion's proposition, and in any case is a published independent result rather than an assumption of the target theorem. The self-citations to [16,50,58,60] are used for the Q=id case and for uniqueness/full-support properties of equilibrium states; they do not assume the projected-pressure results or the almost-sure dimension equalities being proved. No parameter is fitted to data and no known pattern is merely renamed. The main residual risk is the unproved non-invertible extension in Lemma 5.2; an unverified input is a correctness risk, not circularity. A separate possible gap in the final step of Theorem 2(b), where an essential supremum appears where the Hausdorff dimension of a measure needs an essential infimum, would also be a correctness concern rather than a circular reduction.

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

The paper introduces no free parameters: the quantities s_j, dim^Q_aff A, and the exponents in Theorem B are determined by pressure or explicit inequalities, not fitted to data. It introduces no new physical or mathematical entities. The key background assumptions are standard theorems imported from the literature (thermodynamic formalism, proximal representation lemma, Oseledets theorem, integral estimates), plus the domain assumptions in the theorem statements (contraction gap, reductive group for examples).

assumptions (6)
  • standard math Theorem 4: existence, uniqueness and psi-mixing of equilibrium states for submultiplicative and quasi-multiplicative potentials
    Used in Sections 3.2-3.5 to establish equilibrium states of Psi_{U,b} and their mixing properties; cited to [21,31,68].
  • standard math Lemma 3.2: every irreducible representation admits a proximal irreducible representation with comparable norms
    Used in the proof of Proposition 3.1 (bridging lemma) in Section 3.3; borrowed from [12, Lemma 4.13(c)].
  • standard math Lemma 5.2: integral estimate over translation parameters for ||Q Pi_v(i) - Q Pi_v(j)||^{-s}
    Key to the lower bound in Theorem 2(b) (Section 5.2); stated to follow from [8, Section 9.4] and [26, Lemma 2.2].
  • standard math Oseledets multiplicative ergodic theorem for non-invertible cocycles
    Used in Section 3.6 to prove existence of the limit (3.11) and in Section 6.1.3 to construct flags; cited to [66].
  • domain assumption The separation condition max_{i,j: i != j} ||Ai|| + ||Aj|| < 1 is satisfied for the tuples considered in the almost-every statements
    Hypothesis in Theorem A(b), Theorem 2(b), Corollary 2.3; the paper notes it cannot be removed entirely [25,61,75].
  • domain assumption In Theorem 3, G is a real reductive linear algebraic group and wedge^k R^d decomposes as stated
    Hypotheses in Theorem 3(i)-(iii) that enable the algebraic variety construction; not needed for the main Theorem A.

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Pith. "Pith review of Projections of self-affine fractals." pith.science (2026). https://pith.science/paper/RAX2PQFG

@misc{pith2026250204001,
  author       = {Pith},
  title        = {Pith review of: Projections of self-affine fractals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RAX2PQFG}},
  note         = {Machine review of arXiv:2502.04001}
}
read the original abstract

We extend Falconer's 1988 landmark result on the dimensions of self-affine fractals to encompass the dimensions of their projections, showing furthermore that their families of exceptional projections contain algebraic varieties which are preserved by the underlying linear algebraic group. The techniques which we develop allow us to construct examples of additional new phenomena: firstly, we give general examples of equilibrium measures on self-affine fractals which admit non-exact-dimensional projections. Secondly, we construct strongly irreducible self-affine sets which have small sumsets without any arithmetic resonance in their construction.

Figures

Figures reproduced from arXiv: 2502.04001 by the authors.

Figure 1
Figure 1. Two projections of the attractor of an iterated func￾tion system whose linearisation generates a Zariski-dense subsemi￾group of R ∗ SO(2, 2). The two projections are onto two-dimensional isotropic subspaces of R 4 with differing orientations. by Proposition 2.2. Then for every j = 1, . . . , m, Lebesgue a.e. u ∈ (R d ) I we have dimH QXu = tj for Lebesgue a.e. Q ∈ Wtj . Remark 2.4. In general this conclusion cannot … view at source ↗

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Cited by 1 Pith paper

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  1. Projections of self-affine sets onto lines

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    Under strong pinching and strong irreducibility of the linear parts, every line projection of a self-affine set attains the expected dimension; in the plane, strong irreducibility alone suffices.

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