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arxiv: 2506.06851 · v2 · submitted 2025-06-07 · 🧮 math.DS

Weakly separated self-affine carpets

Bal\'azs B\'ar\'any , Levente David This is my paper

Pith reviewed 2026-05-19 10:35 UTC · model grok-4.3

classification 🧮 math.DS
keywords self-affine carpetsHausdorff dimensionbox-counting dimensionweak separation conditioniterated function systemsBarański formulaFeng-Wang formula
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The pith

For self-affine carpets with weakly separated projections, the Hausdorff dimension equals the limit of the Barański formula.

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

The paper focuses on diagonally aligned self-affine carpets whose projections onto the x-axis and y-axis meet the weak separation condition. It proves that the Hausdorff dimension of such a carpet is given exactly by the limit of the Barański formula applied to successive approximations. The box-counting dimension is shown to equal the limit of the Feng-Wang formula taken over the n-fold compositions of the underlying iterated function system. Equivalent expressions for the box-counting dimension are derived, and explicit values are computed for two concrete examples.

Core claim

We show that for diagonally aligned self-affine carpets whose projections to the x- and y-axes satisfy the weak separation condition, the Hausdorff dimension equals the limit of the Barański formula, and the box-counting dimension is the limit of the Feng-Wang formula taken over the n-fold compositions of the IFS. We also prove several equivalent formulas for the box-counting dimension, and derive the dimension values for two examples.

What carries the argument

The weak separation condition imposed on the projections to the x-axis and y-axis, which removes interference from overlaps and lets the dimension formulas reduce to explicit limits.

If this is right

  • The Hausdorff and box-counting dimensions become computable from iterated formulas without separate overlap corrections.
  • Multiple equivalent expressions exist for the box-counting dimension, allowing cross-checks or alternative calculations.
  • Explicit numerical dimension values follow directly for any example satisfying the projection condition.

Where Pith is reading between the lines

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

  • The same separation hypothesis may allow similar limit formulas to hold for self-affine sets that are not strictly carpet-shaped.
  • Numerical truncation of the n-fold compositions offers a practical method to approximate the dimensions from finite data.
  • The results suggest that weak separation on coordinate projections could serve as a testable criterion for dimension formulas in broader classes of affine iterated function systems.

Load-bearing premise

The projections of the self-affine carpet to the x-axis and y-axis satisfy the weak separation condition.

What would settle it

A concrete diagonally aligned self-affine carpet whose projections obey the weak separation condition but whose measured Hausdorff dimension differs from the limit of the Barański formula.

Figures

Figures reproduced from arXiv: 2506.06851 by Bal\'azs B\'ar\'any, Levente David.

Figure 1
Figure 1. Figure 1: Example for application of Theorem 1.1 under the assumptions (C1), (G1) and (W2). [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Example for the application of Theorems 1.1 and 1.3 under the assumption (W1). [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Visualizing the argument for a fixed cylinder and three [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Illustration of a similar argument as in the proof of Lemma 4.8, exept insead of intervals, [PITH_FULL_IMAGE:figures/full_fig_p023_4.png] view at source ↗
read the original abstract

In this paper, we study the Hausdorff and the box-counting dimensions of diagonally aligned self-affine carpets whose projections to the $x$- and $y$-axes satisfy the weak separation condition. In particular, we show that the Hausdorff dimension equals the limit of the Bara\'nski formula, and that the box-counting dimension is the limit of the Feng-Wang formula taken over the $n$-fold compositions of the IFS. We also prove several equivalent formulas for the box-counting dimension, and derive the dimension values for two examples.

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 / 2 minor

Summary. The manuscript studies the Hausdorff and box-counting dimensions of diagonally aligned self-affine carpets whose projections to the x- and y-axes satisfy the weak separation condition. It proves that the Hausdorff dimension equals the limit of the Barański formula and that the box-counting dimension equals the limit of the Feng-Wang formula taken over n-fold compositions of the IFS. Equivalent expressions for the box-counting dimension are derived, and explicit dimension values are computed for two concrete examples.

Significance. If the central claims hold, the work extends dimension theory for self-affine sets by showing that the weak separation condition on projections suffices to equate the dimensions to the indicated limits of the Barański and Feng-Wang expressions. The direct limit arguments over iterated IFS compositions, together with the equivalent box-dimension formulas and worked examples, provide concrete computational tools and avoid parameter-fitting circularities. This strengthens the applicability of existing formulas to a broader class of carpets with controlled overlaps.

minor comments (2)
  1. [Abstract] The abstract states the weak separation condition on projections as the key hypothesis but does not recall its precise definition; adding a one-sentence reminder would improve accessibility for readers outside the immediate subfield.
  2. [Introduction] Notation for the n-fold compositions of the IFS and the associated pressure functions should be introduced with a short display equation in the introduction to make the limit statements easier to parse on first reading.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work on the Hausdorff and box-counting dimensions of diagonally aligned self-affine carpets under the weak separation condition on projections, and for highlighting the significance of the limit expressions and computational tools provided. We appreciate the recommendation of minor revision and note that no specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes equalities between the Hausdorff and box-counting dimensions of diagonally aligned self-affine carpets and the limits of the Barański and Feng-Wang formulas (over n-fold IFS compositions) under the explicit weak separation condition on the projections. These are proved as direct consequences of the projection hypothesis controlling overlaps, using standard limit arguments from dimension theory for iterated function systems. No step reduces a claimed prediction or result to a fitted parameter, self-citation chain, or definitional equivalence by construction; the central claims rest on independent analytic control of the overlaps rather than renaming or smuggling prior results. The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Results rest on standard properties of iterated function systems and the weak separation condition on projections; no free parameters, invented entities, or ad-hoc axioms are visible from the abstract.

axioms (2)
  • standard math Standard properties of iterated function systems generating self-affine sets
    Invoked implicitly as the generating mechanism for the carpets.
  • domain assumption Weak separation condition on axis projections
    The central hypothesis stated in the abstract that enables the dimension formulas.

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Reference graph

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25 extracted references · 25 canonical work pages · 1 internal anchor

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