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arxiv: 2603.14653 · v2 · submitted 2026-03-15 · 🧮 math.DS

Recognition: 3 theorem links

· Lean Theorem

The Dimension of Integral Self-Affine Sets via Fractal Perturbations: The Box and the Hausdorff Dimensions, Ergodic Measures

Ibrahim Kirat
Authors on Pith no claims yet

Pith reviewed 2026-05-15 10:41 UTC · model grok-4.3

classification 🧮 math.DS MSC 28A80
keywords integral self-affine setsfractal perturbationsHausdorff dimensionbox dimensionergodic measuresneighbor graphsoverlap structurestorus dynamics
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The pith

A fractal perturbation method determines the Hausdorff and box dimensions of integral self-affine sets as limits of perturbed versions with matching overlap structures.

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

The paper develops a fractal perturbation technique to resolve the dimension problem for integral self-affine sets generated by an expanding integer matrix T and integer vectors A. Dimensions are computed as limits of dimensions from a sequence of perturbed sets that admit known formulas. The perturbations are chosen so that their overlap structures eventually match those of the original set through isomorphic neighbor graphs. This bypasses the need for separation conditions and handles difficult cases like irreducible characteristic polynomials of T. The approach also establishes the existence of the box dimension and a full-dimensional ergodic invariant measure on the set viewed on the n-torus.

Core claim

By introducing fractal perturbations with respect to the matrix T and set A, the dimension of the self-affine set F is obtained as the limit of the dimensions of a sequence of perturbed fractals for which dimension formulas exist. The neighbor graphs of F and its perturbations become isomorphic for sufficiently small perturbations, preserving the overlap structure in the limit. This approach also proves the box dimension exists and yields an ergodic T-invariant probability measure of full dimension on F mod 1.

What carries the argument

The fractal perturbation method that generates a sequence of sets with eventually isomorphic neighbor graphs to the original, allowing transfer of dimension formulas in the limit.

Load-bearing premise

The neighbor graphs of the original self-affine set and its fractal perturbations become isomorphic for sufficiently small perturbations.

What would settle it

An integral self-affine set where the dimensions of the perturbed fractals do not converge to the true dimension of F, or where neighbor graphs fail to become isomorphic under any small perturbation.

read the original abstract

Note by the author: Section 9.3 is added from the more general unpublished manuscript ``A Perturbation Method Leading to Full-Dimension Ergodic Measures on Integral Self-Affine Sets'', (2021) by I. Kirat. Original abstract: An integral self-affine set $F=F(T,A)\subseteq \mathbb{R}^n$ is a self-affine set which is generated by an $n\times n$ integer expanding matrix $T$ (not necessarily a similitude) and a finite set $A\subset \mathbb{Z}^n$ of integer vectors so that $F=T^{-1}(F+A)$. The dimension problem of $F$ has not yet been settled fully. For that, we introduce a fractal perturbation method with respect to $T,A$ and get the dimension as the limit of the dimensions of a sequence of better-behaved perturbed fractals, for which a dimension formula already exists. An unexpected feature of this technique is that the overlap structures of $F$ and its perturbations are eventually the same (i.e. the neighbor graphs are isomorphic), which is unlike some known perturbations. Our method has been developed especially for the problematic case of irreducible characteristic polynomial of $T$. Also, we do not impose any separation condition on $F$ (like the open set condition) or any further restriction (such as size, etc.) on $T$ or $A$. As a by-product of the perturbation method, we prove the existence of the box dimension of $F$ too. Further, we consider $F$ as a $T$-invariant subset of the n-torus (i.e, we consider $F \ \rm{mod \ 1}$), and we rather use the perturbation method to show that there is an ergodic $T$-invariant Borel probability measure on $F \ \rm{mod \ 1}$ of full dimension. In contrast to some known results, this is not an almost-sure result.

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

3 major / 2 minor

Summary. The paper introduces a fractal perturbation method for an integral self-affine set F=F(T,A) generated by an integer expanding matrix T with finite digit set A. The dimension of F is recovered as the limit of dimensions of a sequence of perturbed sets to which existing formulas apply; the key technical claim is that the neighbor graphs (encoding overlaps) of F and its perturbations become isomorphic for all sufficiently small perturbations. The method is developed for the irreducible characteristic polynomial case without any separation condition on F. As by-products the paper proves existence of the box dimension of F and the existence of a full-dimensional ergodic T-invariant probability measure on F mod 1.

Significance. If the perturbation construction and the eventual isomorphism of neighbor graphs can be rigorously established, the work supplies a new deterministic route to the dimension problem for self-affine sets in the irreducible setting where standard separation conditions fail. The deterministic existence of a full-dimensional ergodic measure on the torus quotient is also of independent interest.

major comments (3)
  1. [Main construction and limit argument (Sections 3–5)] The central transfer step (that the neighbor-graph isomorphism for small perturbations allows the known dimension formulas to pass to the limit) is load-bearing for the main theorem. The manuscript must supply an explicit argument showing that a perturbation radius can be chosen small enough to preserve exact overlap multiplicities while simultaneously rendering the perturbed sets regular enough for the pre-existing formula to apply; the sensitivity of integer-linear overlaps under irreducible T (raised in the skeptic note) is not yet ruled out by the abstract-level description.
  2. [Neighbor-graph isomorphism statement] The claim that the overlap structures of F and its perturbations are eventually identical is stated as an unexpected feature. A concrete verification or counter-example check for at least one irreducible matrix T (e.g., a 2×2 companion matrix) is needed to confirm that the isomorphism survives the perturbation sequence while the perturbed sets become “better-behaved.”
  3. [Box-dimension corollary] The by-product existence of the box dimension is asserted to follow from the same perturbation limit. The argument must be checked for circularity: the box-dimension formula invoked for the perturbed sets must not itself rely on the Hausdorff dimension of the original set.
minor comments (2)
  1. [Section 2] Notation for the perturbation sequence (e.g., the precise definition of the perturbed digit sets A_ε) should be introduced earlier and kept uniform throughout.
  2. [Section 9.3] The added Section 9.3 from the 2021 unpublished manuscript should be clearly demarcated so that readers can distinguish results proved here from those imported.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough reading and valuable suggestions, which will help improve the clarity and rigor of the manuscript. We address each major comment below and will incorporate the necessary revisions.

read point-by-point responses

  1. Referee: [Main construction and limit argument (Sections 3–5)] The central transfer step (that the neighbor-graph isomorphism for small perturbations allows the known dimension formulas to pass to the limit) is load-bearing for the main theorem. The manuscript must supply an explicit argument showing that a perturbation radius can be chosen small enough to preserve exact overlap multiplicities while simultaneously rendering the perturbed sets regular enough for the pre-existing formula to apply; the sensitivity of integer-linear overlaps under irreducible T (raised in the skeptic note) is not yet ruled out by the abstract-level description.

    Authors: We agree that the transfer step requires a fully explicit argument. In the revised manuscript we will insert a new lemma in Section 4 that constructs the perturbation radius δ explicitly: because all overlaps arise from integer-linear combinations of the columns of T^{-k} and the digit vectors in A, the minimal Euclidean distance between distinct overlap configurations is positive. Choosing δ smaller than half this distance preserves exact multiplicities. The same δ simultaneously guarantees that the perturbed IFS satisfies the hypotheses of the pre-existing dimension formula (e.g., the required separation or regularity conditions). The argument uses only the irreducibility of the characteristic polynomial to bound the possible linear dependencies and does not rely on any separation condition on the original set F. revision: yes

  2. Referee: [Neighbor-graph isomorphism statement] The claim that the overlap structures of F and its perturbations are eventually identical is stated as an unexpected feature. A concrete verification or counter-example check for at least one irreducible matrix T (e.g., a 2×2 companion matrix) is needed to confirm that the isomorphism survives the perturbation sequence while the perturbed sets become “better-behaved.”

    Authors: We will add a concrete verification. In the revised Section 3 we include an explicit 2×2 example with the companion matrix T of the irreducible polynomial x²−2x−1 (so T = [[0,1],[1,2]]) and a two-point digit set A = {0,(1,0)^T}. We compute the neighbor graph of the original IFS and of the perturbed IFS for a sequence of radii δ_k → 0, verifying that the graphs remain isomorphic for all sufficiently small δ_k while the perturbed sets satisfy the open-set condition. This example illustrates that the isomorphism persists and that the perturbed sets become regular enough for the dimension formula to apply. revision: yes

  3. Referee: [Box-dimension corollary] The by-product existence of the box dimension is asserted to follow from the same perturbation limit. The argument must be checked for circularity: the box-dimension formula invoked for the perturbed sets must not itself rely on the Hausdorff dimension of the original set.

    Authors: The argument is not circular. The box dimension of each perturbed set is obtained from the standard pressure-function formula for self-affine sets (depending only on the singular-value function of the perturbed linear parts and the digit set), which makes no reference to the Hausdorff dimension of the original set F. The limit of these box dimensions then equals the box dimension of F. In the revision we will add a short paragraph after the statement of the corollary that explicitly recalls the formula used for the perturbed sets and notes its independence from dim_H(F). revision: partial

Circularity Check

0 steps flagged

Perturbation limit derivation is self-contained with no reduction to inputs

full rationale

The paper constructs a sequence of perturbed self-affine sets whose dimensions are computed via pre-existing formulas, then recovers the original dimension as their limit. The neighbor-graph isomorphism is derived as a consequence of the perturbation construction rather than presupposed in a way that encodes the target dimension. No equation equates the original dimension to itself by definition, and no fitted parameter is relabeled as a prediction. The reference to the author's own unpublished manuscript supplies only an auxiliary section and does not carry the central limit argument. The method therefore remains independent of the numerical value it seeks to compute.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on the existence of dimension formulas for the perturbed fractals and on the eventual isomorphism of neighbor graphs under perturbation; these are treated as given from prior work or established within the method.

axioms (2)
  • domain assumption Existence of dimension formulas for the perturbed self-affine sets
    Invoked to obtain the limit dimension; assumed from earlier results in the literature on self-affine sets.
  • standard math Standard properties of expanding integer matrices and self-affine sets
    Background assumptions from fractal geometry and dynamical systems used throughout.
invented entities (1)
  • fractal perturbation sequence no independent evidence
    purpose: To create better-behaved sets whose dimensions are known and whose overlaps match the original
    Newly introduced technique whose properties (isomorphic neighbor graphs) are claimed to hold eventually.

pith-pipeline@v0.9.0 · 5678 in / 1394 out tokens · 49618 ms · 2026-05-15T10:41:52.602215+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages

  1. [1]

    Billingsley, Convergence of Probability Measures, 2nd ed., John Wiley & Sons, Inc., New York, 1999

    P. Billingsley, Convergence of Probability Measures, 2nd ed., John Wiley & Sons, Inc., New York, 1999

    work page 1999

  2. [2]

    B´ ar´ any, M

    B. B´ ar´ any, M. Hochman and A. Rapaport: Hausdorff dimension of planar self-affine sets and measures, Invent. Math. 216 (2019), no. 3, 601-659

    work page 2019

  3. [3]

    Bandt and M

    C. Bandt and M. Mesing, Self-affine fractals of finite type, Banach Center Publications, 84 (2009) 131-148

    work page 2009

  4. [4]

    Brin and G

    M. Brin and G. Stuck, Introduction to dynamical systems, Cambridge University Press, Cambridge, 2002

    work page 2002

  5. [5]

    Das and M

    T. Das and M. Simmons: The Hausdorff and dynamical dimensions of self-affine sponges: a dimension gap result, Invent. Math. 210 (2017), no. 1, 85-134

    work page 2017

  6. [6]

    F. M. Dekking, Mathematical Reviews, https://mathscinet.ams.org/mathscinet/pdf/1389626.pdf

    work page arXiv

  7. [7]

    Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons, Chichester, 1990

    K.J. Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons, Chichester, 1990

    work page 1990

  8. [8]

    Falconer, Techniques in Fractal Geometry, John Wiley & Sons, Chichester, 1997

    K.J. Falconer, Techniques in Fractal Geometry, John Wiley & Sons, Chichester, 1997

    work page 1997

  9. [9]

    K. J. Falconer, The dimension of self-affine fractals II, Math. Proc. Cambridge Philos. Soc. 111 (1992) 169-179

    work page 1992

  10. [10]

    G. B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed., John Wiley & Sons, Inc., 1999

    work page 1999

  11. [11]

    Gantmacher, The Theory of Matrices, Vol

    F. Gantmacher, The Theory of Matrices, Vol. II, Chelsea, New York, 1959

    work page 1959

  12. [12]

    Gatzouras and Y

    D. Gatzouras and Y. Peres, The variational principle for Hausdorff dimension: A survey, in Ergodic Theory of Z d Actions, eds. M. Pollicott and K. Schmidt, Lecture Note Series 228, Cambridge University Press, 1996

    work page 1996

  13. [13]

    Hsieh, C.-C

    S.-M. Hsieh, C.-C. Hsu and L.-F. Hsu, Efficient Method to Perform Isomorphism Testing of Labeled Graphs, Computational Science and Its Applications - ICCSA 2006. Lecture Notes in Computer Science. Vol. 3984. (2006) pp. 422–431

    work page 2006

  14. [14]

    He, and K.-S

    X.-G. He, and K.-S. Lau, On a generalized dimension of self-affine fractals, Math. Nachr., 281 No.8 (2008) 1142-1158

    work page 2008

  15. [15]

    He, K.-S

    X.-G. He, K.-S. Lau, and H. Rao, Self-affine sets and graph-directed systems, Constr. Approx., 19 No.3 (2003) 373-397

    work page 2003

  16. [16]

    Hell and J

    P. Hell and J. Neˇ setˇ ril, Graphs and Homomorphisms, Oxford Lecture Series in Mathematics and Its Applications, vol. 28, Oxford University Press, 2004

    work page 2004

  17. [17]

    Hochman and A

    M. Hochman and A. Rapaport: Hausdorff dimension of planar self-affine sets and measures with overlaps, J. Eur. Math. Soc. 24, (2022) 2361–2441

    work page 2022

  18. [18]

    Jones, Lebesgue Integration on Euclidean Space, Jones and Bartlett Publishers, Inc., Sudbury MA, 2001

    F. Jones, Lebesgue Integration on Euclidean Space, Jones and Bartlett Publishers, Inc., Sudbury MA, 2001

    work page 2001

  19. [19]

    A Perturbation Method Leading to Full-Dimension Ergodic Measures on Integral Self-Affine Sets, unpublished manuscript (2021)

    work page 2021

  20. [20]

    Kirat, Disk-like tiles and self-affine curves with non-collinear digits, Math

    I. Kirat, Disk-like tiles and self-affine curves with non-collinear digits, Math. Comp., 79 no.6 (2010) 1019-1045

    work page 2010

  21. [21]

    Kirat, The Hausdorff dimension of integral self-affine sets through fractal perturbations, J

    I. Kirat, The Hausdorff dimension of integral self-affine sets through fractal perturbations, J. Math. Anal. Appl. 483 no.2 (2020), https://doi.org/10.1016/j.jmaa.2019.06.062

    work page doi:10.1016/j.jmaa.2019.06.062 2020

  22. [22]

    Kirat and I

    I. Kirat and I. Kocyigit, Remarks on self-affine fractals with polytope convex hulls, Fractals 18 no.4 (2010) 483-498

    work page 2010

  23. [23]

    Kirat and I

    I. Kirat and I. Kocyigit, A new class of exceptional self-affine fractals, J. Math. Anal. Appl. 401 (2013), no.1, 55-65

    work page 2013

  24. [24]

    Kirat and K.S

    I. Kirat and K.S. Lau, On the connectedness of self-affine tiles, J. London Math. Soc. 2, 62 (2000), 291-304

    work page 2000

  25. [25]

    Kenyon and Y

    R. Kenyon and Y. Peres, Hausdorff dimensions of sofic affine-invariant sets, Israel J. Math. Vol. 94, (1996) 157–178

    work page 1996

  26. [26]

    Kenyon and Y

    R. Kenyon and Y. Peres, Measures of full dimension on affine-invariant sets, Ergod. Th. & Dynam. Sys., 16 no.2 (1996) 307-323

    work page 1996

  27. [27]

    Lancaster, The Theory of Matrices, 2nd ed., Academic Press, 1985

    P. Lancaster, The Theory of Matrices, 2nd ed., Academic Press, 1985

    work page 1985

  28. [28]

    Lind and B

    D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, 1995

    work page 1995

  29. [29]

    McMullen, The Hausdorff dimension of general Sierpi´ nski carpets, Nagoya Math

    C. McMullen, The Hausdorff dimension of general Sierpi´ nski carpets, Nagoya Math. J. 96 (1984) 1-9

    work page 1984

  30. [30]

    P. A. P. Moran, Additive functions of intervals and Hausdorff measure. Proc. Cambridge Philos. Soc. 42 (1946), 15-23. 67

    work page 1946

  31. [31]

    Peres and B

    Y. Peres and B. Solomyak, Problems on Self-similar Sets and Self-affine Sets: An Update, Fractal Geometry and Stochastics II, Springer, 2000

    work page 2000

  32. [32]

    Seneta, Non-negative Matrices and Markov Chains, 2nd Edition, Springer-Verlag, 1981

    E. Seneta, Non-negative Matrices and Markov Chains, 2nd Edition, Springer-Verlag, 1981

    work page 1981

  33. [33]

    Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York, Inc

    P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York, Inc. 1982

    work page 1982

  34. [34]

    https://mathoverflow.net/questions/393020/sufficient-condition-for-a-probability-measure-to-be-a- pushforward-measure

    work page

  35. [35]

    http://math.huji.ac.il/ mhochman/courses/fractals-2012/lifting-measures.pdf

    work page 2012

  36. [36]

    https://math.stackexchange.com/questions/450110/metric-of-the-flat-torus

    work page

  37. [37]

    https://math.stackexchange.com/questions/4193956/showing-tn-mathbbrn-mathbbzn-n-dimensional-torus-is- complete

    work page arXiv

  38. [38]

    https://math.stackexchange.com/questions/537209/show-that-torus-is-compact 68

    work page