Fibre stability for dominated self-affine sets

Alex Rutar; Roope Anttila

Fibre stability for dominated self-affine sets

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Pith's one-line read For weakly dominated planar self-affine sets, the dimension lost under projection is exactly the largest slice dimension, and the formula holds without separation assumptions.

desk verdict Genuine advance on Fraser's fibre stability question, with an elementary but intricate proof; send it to review. read the letter →

arxiv 2412.06579 v1 pith:VD2HTSHC submitted 2024-12-09 math.DS math.CAmath.MG

Roope Anttila , Alex Rutar This is my paper · ORCID

classification math.DSmath.CAmath.MG MSC 28A8037C4537D30

keywords self-affinesetsAssouaddimensionweaktangentsslicesFurstenbergdirectionsdominationfibrestabilityconformal

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

Self-affine fractals are generated by affine contractions, and a basic question is how much dimension survives when the fractal is projected onto a line. This paper establishes a fibre stability formula for planar self-affine sets whose linear parts are weakly dominated: across every backward Furstenberg direction $V$, the Assouad dimension $\dim_A \pi_{V^\perp}(K)$ of the projection is the same constant $\eta$, and the deficit $\dim_A K - \eta$ equals the largest Hausdorff dimension of a slice of a weak tangent of $K$ perpendicular to $V$. The result matters because it removes the separation and irreducibility hypotheses that all earlier fibre stability theorems needed, and it also settles a previously open question for dominated self-affine sets satisfying strong separation. Under the weak bounded neighbourhood condition the equality transfers to genuine slices of $K$ itself, so the dimension lost in the projection is literally stored in a fibre of the set.

What carries the argument

The engine is Theorem 4.9, an amplification result for slices of weak tangents. Starting from a slice of a weak tangent $E$ in a backward Furstenberg direction, the proof uses a discretized pigeonholing lemma (Corollary 2.4) to locate dyadic scales where the slice has uniformly large branching; this turns a thin tube into a configuration that, after applying a high iterate of a contraction, becomes a coarse microset $A \times B$ with $\dim_B A = \dim_A \pi_{W^\perp}(K)$ and $\dim_B B = \dim_A(\pi_{V^\perp}^{-1}(x)\cap E)$. The geometric lemmas on weakly dominated matrix semigroups keep forward and backward directions separated through a positive angular gap $\delta = \measuredangle(X_F, Y_F) > 0$, which is what lets the pigeonholing survive projection. The product structure is what converts a lower bound on a slice into a lower bound on the Assouad dimension of $K$.

What would settle it

For a weakly dominated planar self-affine IFS, compute both sides of Theorem B when all linear parts are conformal, so $X_F = Y_F$ and $\delta = 0$; if the equality fails, the theorem is false, and if it still holds, the angular gap is not actually necessary. A concrete case is the diagonal carpet of Corollary C with $\alpha = 1/4$, $\beta = 1/2$, two maps, and translations chosen to avoid the grid structure: comparing $\dim_A K$ with $\dim_A \pi(K) + \log(2\cdot (1/2)^s)/\log 4$ for the Frostman dimension $s$ of the projected self-similar measure would settle the formula for that system.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem B. For a weakly dominated planar self-affine IFS with attractor $K$, the map $V \mapsto \dim_A \pi_{V^\perp}(K)$ is constant on the backward Furstenberg directions $X_F$, with value $\eta \le \dim_A K$, and for all $V \in X_F$, $$\max_{E\in\operatorname{Tan}(K)}\max_{x\in \pi_{V^\perp}(E)} \dim_H(\pi_{V^\perp}^{-1}(x)\cap E) = \dim_A K - \eta.$$ If the attractor satisfies the weak bounded neighbourhood condition, then $\dim_A K = \eta + \max_{V\in X_F}\max_{x\in\pi_{V^\perp}(K)}\dim_H(\pi_{V^\perp}^{-1}(x)\cap K)$. The first equality has no separation assumptions; the second needs only a bounded number of overlapping cylinders, a condition weaker than strong separation. The authors interpret this as stability under projections: the largest fibre always stores exactly the dimension lost, and it does so simultaneously in every backward Furstenberg direction.

Load-bearing premise

The load-bearing premise is the positive angular gap $\delta = \measuredangle(X_F, Y_F) > 0$ between backward and forward Furstenberg directions, stated without proof; if the matrix parts were all conformal, the two direction sets would coincide, $\delta = 0$, and Lemma 4.6's separated-projection argument would fail.

Editorial extensions

If this is right

Fibre stability holds for all weakly dominated planar self-affine sets satisfying the weak bounded neighbourhood condition, including strongly separated ones, closing the previously open case of Question 17.5.1 in the cited monograph.
The weak-tangent version of the formula and the upper bound for slices of $K$ require no separation at all, so the dimension deficit is controlled even when cylinders overlap badly.
For non-grid self-affine carpets, the earlier upper bound is upgraded to the equality $\dim_A K = \dim_A \pi(K) + \log(m\beta^s)/\log(1/\alpha)$.
Weakly dominated irreducible self-affine sets satisfy a dichotomy for conformal Assouad dimension: sets with Assouad dimension below 1 have conformal Assouad dimension 0, and sets with Assouad dimension at least 1 are minimal.

Reading between the lines

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

The proof's reliance on $\delta>0$ suggests a testable technical question: whether the theorem survives when the angular gap degenerates, since the paper supplies no example where equality actually fails in that regime.
The pigeonholing amplification is likely to transfer to higher-dimensional self-affine sets, where a slice is replaced by a $k$-dimensional fibre; the same product-microset strategy would need a multidimensional version of the branching lemma.
The weak bounded neighbourhood condition is sufficient for pulling weak-tangent slices back to $K$, but Question 1.3 leaves open whether the Assouad-dimension version of the slice equality holds without any separation assumption; the absence of a counterexample in the paper makes this a natural next test case.

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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

Two things you should know. This is the first fibre stability result for weakly dominated planar self-affine sets with no separation and no irreducibility assumptions, which genuinely advances Fraser's Question 17.5.1. And the proof is elementary in the good sense: pigeonholing plus Furstenberg's microset construction, with no heavy projection theorems.

What is new: Theorem B gives the slice formula for weak tangents uniformly over all backward Furstenberg directions, and under strong separation or the weak bounded neighbourhood condition it upgrades to slices of K itself. That uniformity is genuinely stronger than earlier results. The core contribution is Theorem 4.9, an amplification argument showing that a slice of a weak tangent of Assouad dimension beta forces a product A x B to be a coarse microset of K, with dim_B A = dim_A pi(K) and dim_B B = beta. The diagonal case in the appendix is a useful service to readers.

Credit where earned: the paper gives detailed proofs of the main claims, including the geometric lemmas. Proposition A is a clean, self-contained proof of a result that the field seems to have missed until now. The applications are real: Corollary C removes the grid structure from the Fraser-Jordan carpet result, and Corollary D gives a conformal Assouad dimension dichotomy. The citation pattern looks honest; several cited lemmas come from the authors' own earlier work, but they are standard in the area and the reliance is transparent.

Soft spots. The proof of Theorem 4.9 is intricate and I did not verify every pigeonholing line. There are some repairable bookkeeping slips: subscript issues around equation (4.9) and an informal sign normalization before Lemma 4.2. The reader worried about the positive angular gap delta = angle(XF, YF) used in Lemma 4.6; I checked this concern and it does not land. Proposition 3.7 plus Lemma 3.1.5 from [ABK24] gives YF inside the interior of the multicone and XF outside it, and both sets are compact, so delta > 0. The conformal-only case where delta could vanish is excluded by the requirement that the hyperbolic part Ah be non-empty. A minor worry is that the paper leans on the preprint [BKY21+] for some lemmas, but those lemmas are known in the area. The WBNC condition is introduced only to push weak tangents back to K, and the paper is explicit about its role.

Who this is for: anyone working on self-affine geometry, projections and slices, or Assouad dimension. It deserves a serious referee, and the referee should spend most effort on Theorem 4.9. My recommendation: engage with it and send it to review. I would accept with confidence that the theorem is new and the strategy sound, even if a few details need polishing.

Referee Report

0 major / 5 minor

Summary. The paper proves fibre stability for weakly dominated planar self-affine sets. Theorem B states that for every backward Furstenberg direction V, the difference dim_A K − η, where η is the constant value of V ↦ dim_A π_{V⊥}(K) on the backward Furstenberg set X_F, equals the maximal Hausdorff dimension of a slice of a weak tangent of K in direction V⊥. Under the weak bounded neighbourhood condition, the same equality is obtained for slices of K itself, and applications are given to self-affine carpets and to the conformal Assouad dimension of irreducible weakly dominated self-affine sets. The proof is self-contained and avoids the deep projection theorems used in earlier work, relying instead on a pigeonholing argument that amplifies large slices of weak tangents into product-like coarse microsets.

Significance. If the result is correct, it resolves a question of Fraser for dominated self-affine sets in full generality, removing the irreducibility and projection-geometry assumptions that appear in all previous work. The theorem extends known results for diagonal carpets and for strongly irreducible systems, and the no-separation part is new even for diagonal IFS. The paper is also methodologically valuable: the proof is essentially self-contained, gives a short elementary proof of a Furstenberg-type microset lemma (Proposition A), and includes a separate appendix for diagonal systems. The applications to tube dimensions and to the conformal Assouad dimension are concrete and nontrivial.

minor comments (5)

[Theorem 4.1] The statement has the inner supremum over x ∈ π_{V⊥}(K), but the proof and Theorem 4.9 show that the supremum should be over x ∈ π_{V⊥}(E), with E ranging over Tan(K). The subscript K appears to be a typo, since the displayed proof of Theorem 4.1 uses x ∈ π_{V⊥}(E).
[Proof of Theorem 4.9 (after Eq. (4.1))] The assertion that ∥A_j x/∥A_j∥ − π_W^Y x∥ < R/m for all x ∈ R^2 is impossible as written, because the left-hand side is unbounded. It should be restricted to a bounded set, for example x ∈ K − K, and then it follows from uniform convergence on compact sets. This is the interpretation needed for (4.2), so the error is a repairable typo rather than a gap.
[Lemma 4.6] The positivity δ = ∡(X_F, Y_F) > 0 is asserted without proof. A sentence explaining that this follows from Proposition 3.7 and Lemma 3.1(5) would help: for a weakly dominated tuple with A_h non-empty, Y_F ⊂ C° and X_F ⊂ RP^1 \ C for the strongly invariant multicone C, so the two compact sets are disjoint and hence positively separated.
[Corollary 5.4] In the second displayed equation, the final supremum is written as dim_A(π^{-1}_{V⊥}(x) ∩ F) with F not defined in that context; it should be ∩ E, with E ranging over Tan(K), matching the first supremum.
[Theorem B and related displays] There are several subscript/notation slips: in the final display of Theorem B, “dimH(πV ⊥(x)−1 ∩ K)” should be dim_H(π^{-1}_{V⊥}(x) ∩ K); similarly, in Proposition 1.1, “dimH A ≤ η” should be “dimH E ≤ η”.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem B is a new derivation from first-principles pigeonholing plus independent cited results, with no fitted parameter renamed as a prediction.

full rationale

The central claim, Theorem B, is derived from Proposition A, Theorem 4.1/4.9, and Corollaries 5.1–5.4. The proof of Theorem 4.9 constructs a coarse microset A × B via Lemma 2.3 (Furstenberg pigeonholing), Lemma 4.6 (separation under projection), and the weak-domination structure from Section 3. No parameter is fitted to the target quantity: the constants M, m, r1, R′ are chosen by pigeonholing and dyadic covering estimates, and the conclusion dim_A K ≥ η + β is not assumed. The paper's cited results, including [ABK24, Lemma 3.2] and [BKM20], are prior theorems with independent proofs; they are used as tools, not as assertions of the theorem being proved. The only possible concern, the positivity of δ = ∡(XF, YF) in Lemma 4.6, is justified by the strongly invariant multicone in Proposition 3.7 together with Lemma 3.1(5), which places YF in C° and XF in RP^1 \ C; since these are disjoint compact sets, δ > 0. The paper does not rename a known empirical pattern, does not import a uniqueness theorem from the authors, and does not make a prediction that reduces to an input by construction. The counterexample in Proposition 1.2 is cited from [FR24] only to show sharpness, not to establish the main theorem. Therefore no circular step is present.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper's central claim rests on established concepts in dimension theory and matrix cocycles: weak tangents, Assouad dimension, Furstenberg directions, and the cited lemmas on dominated matrix products. No new entities or fitted numerical parameters are introduced. The main original content is the pigeonholing amplification argument, which builds on Furstenberg's microset construction.

assumptions (5)

domain assumption Weak domination of the matrix tuple A (Definition 3.3): A splits into a strongly conformal part Ae and a dominated part Ah with a strongly invariant multicone C such that A(C) = C for A in Ae.
This is the class of IFSs for which the theorem is claimed; the proof proceeds under this assumption and it is not derived.
domain assumption Positive angular separation δ = ∡(XF, YF) > 0 between backward and forward Furstenberg direction sets.
Asserted without proof in Lemma 4.6 and Theorem 4.9. It follows from the strongly invariant multicone in Definition 3.3, but the paper does not demonstrate the implication. If this separation failed, Lemma 4.6 would not hold.
standard math Furstenberg's microset theorem: for a compact F with dim_A F = η, some weak tangent E has H^η_∞(E) ≥ 1.
Cited as [Fur08, Proposition 1.1]; used in Proposition A and Corollary 5.1 to pass from box dimension of slices to Hausdorff dimension.
standard math Standard matrix semigroup lemmas: [ABK24, Lemmas 2.2-2.3] (Furstenberg direction limits and equivariance for dominated tuples), [BKY21+, Lemma 2.8] (singular value comparison), [BKM20, Theorem 2.1, Lemma 3.7, Corollary 2.5] (almost additivity and weak domination structure).
Used in Section 3 to show the Furstenberg directions of the weakly dominated tuple are those of a dominated tuple and to obtain Lemma 3.8.
standard math Orponen's strong projection theorem for Assouad dimension [Orp21], the positive Hausdorff dimension of XF for strongly irreducible semigroups [BL85, Corollary VI.4.2], and [BKY21+, Lemma 2.7] that weak domination plus irreducibility implies strong irreducibility.
Used only in the proof of Corollary D, not in the main slicing theorem.

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Pith. "Pith review of Fibre stability for dominated self-affine sets." pith.science (2026). https://pith.science/paper/VD2HTSHC

@misc{pith2026241206579,
  author       = {Pith},
  title        = {Pith review of: Fibre stability for dominated self-affine sets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VD2HTSHC}},
  note         = {Machine review of arXiv:2412.06579}
}

abstract

Let $K$ be a planar self-affine set. Assuming a weak domination condition on the matrix parts, we prove for all backward Furstenberg directions $V$ that $$\max_{E\in\operatorname{Tan}(K)} \max_{x\in \pi_{V^\bot}(E)} \operatorname{dim_H} (\pi_{V^\bot}^{-1}(x)\cap E) = \operatorname{dim_A} K - \operatorname{dim_A} \pi_{V^\bot}(K).$$ Here, $\operatorname{Tan}(K)$ denotes the space of weak tangents of $K$. Unlike previous work on this topic, we require no separation or irreducibility assumptions. However, if in addition the strong separation condition holds, then there exists a $V\in X_F$ so that $$\max_{x\in \pi_{V^\bot}(K)} \operatorname{dim_H} (\pi_{V^\bot}^{-1}(x)\cap K) = \operatorname{dim_A} K - \operatorname{dim_A} \pi_{V^\bot}(K).$$ Our key innovation is an amplification result for slices of weak tangents via pigeonholing arguments.

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Forward citations

Cited by 1 Pith paper

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Reference graph

Works this paper leans on

7 extracted references · 4 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Anttila, B

[ABK24] R. Anttila, B. Bárány, and A. Käenmäki. Slices of the Takagi function. Ergodic Theory Dynam. Systems 44 (2024), 2361–2398. doi:10.1017/etds.2023.117. [BHR19] B. Bárány, M. Hochman, and A. Rapaport. Hausdorff dimension of planar self-affine sets and measures. Invent. Math. 216 (2019), 601–659. zbl:1414.28014. [BKM20] B. Bárány, A. Käenmäki, and I. ...

work page arXiv 2024

[2] [33]

[FHO+15] J

zbl:07846313. [FHO+15] J. M. Fraser, A. M. Henderson, E. J. Olson, and J. C. Robinson. On the Assouad dimension of self-similar sets with overlaps. Adv. Math. 273 (2015), 188–214. zbl:1317.28014. [Fra14] J. M. Fraser. Assouad type dimensions and homogeneity of fractals. Trans. Amer. Math. Soc. 366 (2014), 6687–6733. zbl:1305.28021. [Fra20] J. M. Fraser. A...

work page arXiv 2015

[3] [1985]

[DOW24] D

zbl:0572.60001. [DOW24] D. D ˛ abrowski, T. Orponen, and H. Wang. How much can heavy lines cover? J. Lond. Math. Soc. 109 (2024),

work page arXiv 2024

[4] [1995]

[Orp21] T

zbl:0819.28004. [Orp21] T. Orponen. On the Assouad dimension of projections. Proc. Lond. Math. Soc. 122 (2021), 317–351. zbl:1465.28008. [OS23] T. Orponen and P . Shmerkin. On the Hausdorff dimension of Furstenberg sets and orthogonal projections in the plane. Duke Math. J. 172 (2023), 3559–3632. zbl:07810678. [RW23+] K. Ren and H. Wang. Furstenberg sets ...

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[5] [2010]

[Mar54] J

zbl:1201.30002. [Mar54] J. M. Marstrand. Some fundamental geometrical properties of plane sets of fractional dimensions. Proc. Lond. Math. Soc. 4 (1954), 257–302. zbl:0056.05504. [Mat95] P . Mattila. Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability. Vol

arXiv 1954

[6] [2017]

[BG09] J

zbl:1390.28012. [BG09] J. Bochi and N. Gourmelon. Some characterizations of domination. Math. Z. 263 (2009), 221–231. zbl:1181.37032. [BL85] P . Bougerol and J. Lacroix. Products of random matrices with applications to Schrödinger operators. Vol

work page arXiv 2009

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[FJ17] J

zbl:1467.28001. [FJ17] J. M. Fraser and T. Jordan. The Assouad dimension of self-affine carpets with no grid structure. Proc. Amer. Math. Soc. 145 (2017), 4905–4918. zbl:1371.28016. 38 R OOPE ANTTILA & ALEX RUTAR [FR24] J. M. Fraser and A. Rutar. Assouad-type dimensions of overlapping self-affine sets. Ann. Fenn. Math. 49 (2024), 3–21. zbl:07808129. [Fur0...

arXiv 2017