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arxiv: 2412.02123 · v4 · submitted 2024-12-03 · 🧮 math.CA · math.MG

Self-embedding similitudes of Bedford-McMullen carpets with dependent ratios

Jian-Ci Xiao This is my paper

Pith reviewed 2026-05-23 08:36 UTC · model grok-4.3

classification 🧮 math.CA math.MG
keywords Bedford-McMullen carpetself-embedding similitudeoblique embeddingcontraction ratiologarithmic commensurabilitygeneralized Sierpiński carpetstrong separation condition
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The pith

Non-degenerate Bedford-McMullen carpets admit no oblique self-embedding similitudes.

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

The paper proves that if a similitude maps a non-degenerate Bedford-McMullen carpet into itself, then the image of the x-axis under that map must lie parallel to one of the coordinate axes. This axis-alignment condition immediately yields a logarithmic commensurability statement for the contraction ratios appearing in the carpet's iterated function system. The argument completes an earlier analysis that had covered only the multiplicatively independent case and supplies a new proof route that does not examine tangent sets. In the purely self-similar setting the paper also exhibits a generalized Sierpiński carpet that does admit an oblique reflectional self-embedding, and it shows that any oblique rotational self-embedding on a strongly separated generalized Sierpiński carpet must have rotation-angle tangent equal to plus or minus one.

Core claim

Any similitude sending a non-degenerate Bedford-McMullen carpet into itself has the property that the image of the x-axis is parallel to a principal axis. This forces logarithmic commensurability of the carpet's contraction ratios.

What carries the argument

The non-obliqueness property of self-embedding similitudes, which forces the transformed x-axis to align with a coordinate axis.

If this is right

  • The logarithms of the contraction ratios of any self-embedding must be commensurable.
  • The classification of self-embeddings is now complete for both dependent and independent ratio cases.
  • Oblique rotational self-embeddings of strongly separated generalized Sierpiński carpets are restricted to angles whose tangent equals ±1.
  • A proof of non-obliqueness is available that does not rely on tangent-set analysis.

Where Pith is reading between the lines

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

  • The axis-alignment requirement may impose similar rigidity on other families of self-affine sets.
  • The existence of an obliquely symmetric generalized Sierpiński carpet indicates that self-similarity alone permits more geometric flexibility than the Bedford-McMullen structure allows.
  • It would be natural to test whether the same non-obliqueness conclusion holds for additional classes of planar self-affine fractals.

Load-bearing premise

The carpet satisfies the non-degeneracy condition that keeps it from collapsing to lower dimension or trivial alignment.

What would settle it

An explicit non-degenerate Bedford-McMullen carpet together with an oblique similitude that maps the carpet to itself while sending the x-axis to a line not parallel to either coordinate axis.

Figures

Figures reproduced from arXiv: 2412.02123 by Jian-Ci Xiao.

Figure 1
Figure 1. Figure 1: An illustration of the local behavior of Case 2 By Lemma 2.3(1), [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The evolution from Rk to f(Rk) to ϕ −1 Qk f(Rk) (color online) Write eak := ϕ −1 Qk f(ak). Since f(ak) ∈ Qk, eak ∈ [0, 1]2 . Fix a large k (will be specified later) so that mk−p nk−p ·max{u −1 , u} < λ and |ϕ −1 Qk f(ℓ k )| ≤ 2n −p √ λ 1+u2 (recall (3.2)). Let us truncate the parallelogram ϕ −1 Qk f(Rk) as S t∈Z Eet , where Eet := ϕ −1 Qk f(Rk) ∩ {x ∈ R 2 : π1(x) ∈ [−t, −t + 1]}. Note that for every t, |π2… view at source ↗
Figure 3
Figure 3. Figure 3: ϕ −1 Qk f(ℓ k ) + ξ0 is contained in the hole U + e (color online) Since ξ0 + ϕ −1 Qk f(ℓ k ) is a translated copy of ϕ −1 Qk f(ℓ k ) and is contained in ϕ −1 Qk f(Rk), f −1ϕQk (ξ0) + ℓ k is a translated copy of ℓ k and is contained in Rk. Since ℓ k is the bottom edge of the level-k rectangle Rk, ℓ k ∩ ϕRk (K) = ϕRk (K0 ). So there is y0 ∈ [0, 1] such that (f −1ϕQk (ξ0) + ℓ k ) ∩ ϕRk (K) = ϕRk (Ky0 × {y0})… view at source ↗
Figure 4
Figure 4. Figure 4: Length of horizontal slices of ϕ −1 Qk f(Rk) If Condition A holds for t0 = 1 and some suitable j0 then we are done. Otherwise, there exists some j∗ such that Ee1 ∩ (R × {j∗ m }) 6= ∅. Since |ϕ −1 Qk f(Rk) ∩ (R × {j∗ m })| ≤ Cn−pλ, picking k large at the beginning, it is not hard to check that Condition A holds for t∗ and j∗, where t∗ := min{t ≥ 2 : Eet ∩ (R × {j∗ m }) = ∅}. Remark 3.3. Note that in the pro… view at source ↗
Figure 5
Figure 5. Figure 5: An illustration of ea,eb, g(Rt) and g(R′ t ), where (Kz + c) × R is supported in the shaded region. (color online) are −1 ≤ c ′ t , γ′ t ≤ 1 and −1 ≤ d ′ t ≤ 0 such that for all t ≥ k0, (4.12) x ′ = (gϕRt ) −1 (ea) + c ′ t d ′ t  = (gϕR′ t ) −1 (eb) + γ ′ t η ′ t  . Fix a large t ≥ k0 (will be specified later). Note that dt − d ′ t = π2(x − x ′ ) = max π2(K) − min π2(K) =: β > 0. So at least one of dt … view at source ↗
Figure 6
Figure 6. Figure 6: A carpet allowing oblique self-embeddings [PITH_FULL_IMAGE:figures/full_fig_p024_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The rotation effect of f (color online) Since ϕR(ℓt) is parallel to ϕR′(ℓt) and ϕR′, ϕR involve no rotations nor reflections, f sends a line parallel to ℓt ′ to another line parallel to ℓt . Without loss of generality, assume that t ′ < t and v1, . . . , vp are in counterclockwise order. Then a simple geometric observation (see [PITH_FULL_IMAGE:figures/full_fig_p027_7.png] view at source ↗
read the original abstract

We prove that any non-degenerate Bedford-McMullen carpet does not admit oblique self-embedding similitudes; that is, if $f$ is a similitude sending the carpet into itself, then the image of the $x$-axis under $f$ must be parallel to one of the principal axes. This result leads to a logarithmic commensurability result on the contraction ratios of such embeddings, completing a previous study by Algom and Hochman [Ergod. Th. & Dynam. Sys. 39 (2019), 577-603] on Bedford-McMullen carpets generated by multiplicatively independent exponents. Our approach also provides a new proof of their non-obliqueness statement that avoids analyzing the tangent sets. For the self-similar case, however, we construct a generalized Sierpi\'nski carpet that is symmetric with respect to an appropriate oblique line and hence admits a reflectional oblique self-embedding. As a complement, we prove that if a generalized Sierpi\'nski carpet satisfies the strong separation condition and permits an oblique rotational self-embedding similitude, then the tangent of the rotation angle takes values $\pm 1$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that any non-degenerate Bedford-McMullen carpet admits no oblique self-embedding similitudes: if f is a similitude with f(K)⊂K then the image of the x-axis under f is parallel to a coordinate axis. This yields a logarithmic commensurability statement on the contraction ratios of such embeddings, completing the Algom-Hochman analysis for multiplicatively independent exponents. The proof avoids tangent-set analysis. For the self-similar case the authors construct a generalized Sierpiński carpet symmetric across an oblique line (hence admitting an oblique reflection) and prove that any SSC generalized Sierpiński carpet admitting an oblique rotational self-embedding similitude must have rotation angle θ satisfying tan θ=±1.

Significance. The non-obliqueness theorem and the derived commensurability result close an open case for Bedford-McMullen carpets with dependent ratios. The new proof technique that bypasses tangent sets is a methodological contribution. The complementary construction demonstrates that the non-obliqueness statement is sharp in the self-similar setting, while the tan θ=±1 restriction supplies a clean necessary condition under SSC. These results are directly relevant to the study of self-similar sets and their symmetries.

minor comments (3)
  1. §1: the non-degeneracy hypothesis is stated only in the abstract and the main theorem; a precise definition (e.g., in terms of the number of rectangles per row/column) should appear in the introduction or preliminaries so that the scope is immediately clear.
  2. The statement of the commensurability corollary (presumably after the main theorem) should include an explicit formula relating log r_i and log s_j rather than leaving the relation implicit.
  3. Figure 1 (or the first illustrative figure of the generalized Sierpiński carpet) would benefit from an overlaid oblique line of symmetry to make the construction visually immediate.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the recognition of its contributions to the study of self-similar sets and symmetries, and the recommendation for minor revision. We are pleased that the non-obliqueness theorem, the commensurability result, the new proof technique, the complementary construction, and the tan θ=±1 restriction are viewed as relevant and sharp.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper is a pure-mathematics proof establishing non-obliqueness of self-embedding similitudes for non-degenerate Bedford-McMullen carpets and deriving a logarithmic commensurability statement. The central argument relies on geometric properties of similitudes and the carpet construction under the stated non-degeneracy hypothesis; this hypothesis is an external premise, not derived from the conclusion. The work cites Algom-Hochman (different authors) only for context and supplies an independent proof that avoids their tangent-set method. No equations reduce a claimed prediction to a fitted input, no uniqueness theorem is imported from the present authors' prior work, and no ansatz is smuggled via self-citation. The complementary construction for generalized Sierpiński carpets is presented as a sharpness example, not as part of the main derivation chain. The derivation is therefore self-contained against external geometric benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on the standard Euclidean geometry of similitudes and the conventional definition of Bedford-McMullen carpets; no free parameters or invented entities are introduced.

axioms (2)
  • standard math Similitudes in the Euclidean plane preserve angles and scale all distances by the same factor.
    Used to deduce that the image of the x-axis must remain axis-parallel if the mapping preserves the carpet.
  • domain assumption Non-degeneracy of a Bedford-McMullen carpet excludes lower-dimensional or trivially aligned constructions.
    Explicitly required for the non-obliqueness statement in the abstract.

pith-pipeline@v0.9.0 · 5731 in / 1473 out tokens · 54125 ms · 2026-05-23T08:36:14.565269+00:00 · methodology

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Reference graph

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