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Crossing lines carry hidden dimension that classical fractals miss

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2026-07-10 04:24 UTC pith:EGNUKADO

load-bearing objection New pointwise directional dimension theory with a load-bearing shadowing lemma that holds up under scrutiny the 1 major comments →

arxiv 2607.08612 v1 pith:EGNUKADO submitted 2026-07-09 math.MG math.CAmath.FAmath.GN

Point-dimension theory (part II): The point-cross dimension

Nadir Maaroufi This is my paper
classification math.MG math.CAmath.FAmath.GN
keywords dimensiondirectionallocalpoint-crossdirectionstheorychannelscomplexity
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper introduces the Point-Cross Dimension, a new point-by-point invariant for subsets of Euclidean space that measures how many independent directional channels coexist at a single point. Classical fractal dimensions—Hausdorff, box, packing, Assouad—obey a max-rule on unions: the dimension of two intersecting sets is just the larger of the two. This erases the local interaction at a crossing. The Point-Cross Dimension is built to recover that interaction. At a point where two transverse line segments meet, it returns 2 rather than 1, because two genuinely independent directions are simultaneously active. The construction proceeds in three layers. First, a point-vector dimension counts exact straight-segment directions issuing from the point. Second, a point-tangential dimension replaces exact segments by Bouligand tangent directions, capturing asymptotic approach directions even when no straight segment is present. Third, the full Point-Cross Dimension weights each effective tangent direction by the point-extended box complexity detectable along admissible Lipschitz probes approaching the point in that direction, then sums these weights over projectively independent directions. The result is an invariant satisfying dim_Pvec ≤ dim_× ≤ dim_Ptan, agreeing with manifold dimension on smooth submanifolds, recovering the incident-edge rank on geometric graphs, and invariant under local C^1-diffeomorphisms and closure. The paper then computes the invariant on oscillatory germs, fractal coordinate frames, Sierpiński-type carpets, and Cantor dusts, and establishes comparison principles showing when directional richness forces dispersive largeness and when it does not.

Core claim

The Point-Cross Dimension dim_×{x}A is defined by assigning to each effective Bouligand tangent direction v a directional contribution θ^A_x(v) = sup over admissible Lipschitz probes γ of the point-extended box dimension of A ∩ Γ_γ, then aggregating: dim_×{x}A = sup over projectively independent families (ξ_1,…,ξ_m) of Σ θ^A_x(ξ_i). This invariant is squeezed between the point-vector dimension (exact segment rank) and the point-tangential dimension (Bouligand tangent span rank), equals k on C^1 k-dimensional submanifolds, equals the linear rank of incident edge directions on geometric graphs, is invariant under closure and local C^1-diffeomorphisms, and—crucially—exceeds the classical max of

What carries the argument

The directional contribution θ^A_x(v) is the central object. It is defined as the supremum, over all admissible Lipschitz probes γ approaching x with limiting direction v, of the point-extended box dimension dim_Pbox{x}(A ∩ Γ_γ). Each probe is an injective Lipschitz curve whose image has box dimension at most 1, so each direction contributes at most 1. The Point-Cross Dimension then sums these per-direction weights over projectively independent effective directions. The Shadowing Lemma (Lemma 3.35) is the load-bearing technical step for closure invariance: it constructs a bi-Lipschitz ambient homeomorphism supported in disjoint balls around points of the closure trace, moving them onto an ad

Load-bearing premise

The closure invariance of the directional contribution relies on a Shadowing Lemma that constructs a bi-Lipschitz homeomorphism across infinitely many disjoint balls to transfer box-dimensional mass from closure traces back to the original set. If the bi-Lipschitz constants accumulate badly across infinitely many disjoint balls, the point-extended box exponent might not survive the transfer, which would compromise the well-definedness of the theory on closed germs.

What would settle it

Find a closed set A ⊂ R^n and a point x ∈ A where a direction v is an effective Bouligand direction and the closure trace Ā ∩ Γ_γ along some probe has point-extended box dimension s > 0, but where no admissible probe meeting A itself can recover dimension s—i.e., where the Shadowing Lemma's bi-Lipschitz construction degrades the separated-set cardinality so badly that the box exponent drops. This would break θ^A_x(v) = θ^{Ā}_x(v) and hence closure invariance of dim_×.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 10 minor

Summary. The paper introduces the Point-Cross Dimension, a new pointwise invariant measuring the directional organization of a set at a single point. The construction proceeds in three layers: (1) the point-vector dimension dim_Pvec, counting exact local segment directions; (2) the point-tangential dimension dim_Ptan, counting asymptotically accessible Bouligand tangent directions; (3) the Point-Cross dimension dim_×, which weights effective projective directions by the point-extended box complexity detected along admissible Lipschitz probes and aggregates over projectively independent channels. The main structural results establish the hierarchy dim_Pvec ≤ dim_× ≤ dim_Ptan, C^1-diffeomorphism invariance, closure invariance, a Grassmann-type union formula, and calibration on smooth submanifolds and geometric graphs. The paper also introduces a combined Point-Box-Cross dimension dim_⊠ = max(dim_Pbox, dim_×). Model examples include oscillatory germs, fractal coordinate frames, and Sierpiński-type carpets. The final part develops comparison principles between directional and dispersive layers.

Significance. The paper addresses a genuine gap in local dimension theory: classical isotropic dimensions (Hausdorff, box, packing, Assouad) obey a max-rule on finite unions and thus collapse the directional richness present at crossing points. The n-axis frame in R^n carrying pointwise dimension 1 everywhere under classical dimensions is a compelling motivating example. The graded per-direction weight θ_A_x(ξ) ∈ [0,1], combined with aggregation over projectively independent directions, is a novel construction that appears to differ from existing directional notions (tangential dimensions of Guido–Isola, direction sets of Koike–Paunescu, Assouad spectra, decomposability bundles of Alberti–Marchese) in the precise sense summarized in Table 3.1. The C^1-invariance (Theorem 4.7) and the smooth calibration (Proposition 3.51) are clean results. The collapse on geometric graphs (Proposition 3.56) serves as a useful calibration check. The comparison with existing notions in Section 3.7 is thorough and helps position the contribution.

major comments (1)
  1. The Shadowing Lemma (Lemma 3.35) is the load-bearing technical step for closure invariance (Proposition 3.37), which in turn underpins the well-definedness of the theory on closed germs. The construction patches countably many bi-Lipschitz maps H_i on pairwise disjoint balls B(p_i, r_i). The global Lipschitz estimate Lip(H) ≤ 2 is argued via decomposition of segments into subsegments and monotone convergence. While the argument appears sound on inspection, the infinite gluing across countably many balls is subtle enough that a brief explicit verification of the summation argument—perhaps as a dedicated paragraph or a reference to a standard gluing lemma for bi-Lipschitz maps on disjoint supports—would strengthen the paper. Specifically, the claim that the segment [z,w] intersects at most countably many disjoint open balls, producing subsegments whose total inflation is bounded by 2||z-w−
minor comments (10)
  1. Sections 1.1–1.3 (philosophical, mathematical, and historical motivations) span approximately 5 pages. While the author explicitly states that the phenomenological analysis is not foundational for the mathematics, the length is disproportionate relative to the technical content. Consider condensing to 2–3 pages, retaining the key motivating examples (Figures 1.1, 1.2).
  2. The Lebesgue quotation in French is untranslated. A brief English paraphrase or translation would aid readers unfamiliar with French.
  3. In Definition 3.2, the condition A ∩ γ((0,η]) ≠ ∅ for every η ∈ (0,δ] is the recurrence condition. This is clear, but the term 'recurrence' is not introduced before its use in the proof of Proposition 3.4. A brief gloss when first used would help.
  4. Remark 3.8 discusses the orientation convention (v vs −v) and the passage from spherical to projective directions. The distinction between oriented and projective formulations recurs throughout Section 3. A summary diagram or a consolidated remark collecting all conventions would improve readability.
  5. In Proposition 2.41, the Grassmann formula for dim_Ptan on unions is exact, but Remark 2.43 notes that the overlap term is not dim_Ptan{x}(A∩B). This is an important subtlety. Consider flagging it more prominently, perhaps with a labeled cautionary example.
  6. The notation θ^A_x(v) uses a superscript for the set A, which can be confused with exponentiation. Consider θ_A(x,v) or θ(x,v;A).
  7. Table 3.1 is helpful but could benefit from a column indicating whether each notion is defined pointwise (at every point) or almost everywhere, since this is a key distinction for the decomposability bundle row.
  8. The paper references 'Proposition 5 in [25]' and 'Corollary 2 in [25]' for upper semicontinuity of dim_Pbox (Remark 3.23) and 'Remark 7 in [25]' for locality. Since these are load-bearing properties, brief restatements of the relevant statements would make the paper more self-contained.
  9. In Example 3.10(2), the computation of Eff_{(0,0)}(Γ_{α,β}) for the three regimes (α>1, α=1, 0<α<1) is detailed but the role of β (oscillatory frequency) is mentioned only at the end. Stating upfront that β does not affect the directional support would orient the reader.
  10. The bibliography appears to use a non-standard format. Ensure consistency with the journal's citation style.

Circularity Check

0 steps flagged

No circularity: the Point-Cross dimension is defined from independent ingredients and its hierarchy properties are proved, not assumed.

full rationale

The paper defines dim×{x}A as the supremum of sums of directional contributions θA_x(ξ) over projectively independent effective directions, where each θA_x(ξ) is defined via the Point-Extended Box Dimension (dimPbox) of traces along admissible Lipschitz probes. The Point-Extended Box Dimension is cited from the author's prior work [24, 25], but it is used as an independent input tool—a local box dimension—rather than being defined in terms of dim× itself. The three-layer hierarchy (dimPvec ≤ dim× ≤ dimPtan) is established by proof: the lower bound (Proposition 3.48) follows because exact local directions yield straight probes with full weight 1; the upper bound (Proposition 3.47) follows because each directional contribution is at most 1 and at most dimPtan independent directions exist. The calibration results (smooth submanifolds give dim× = k in Proposition 3.51; geometric graphs give dim× = incident-edge rank in Proposition 3.56) are proved from the definitions, not assumed. The closure invariance (Proposition 3.37) relies on the Shadowing Lemma (Lemma 3.35), which is a genuine construction using bi-Lipschitz ambient homeomorphisms—not a circular restatement. The C¹-invariance (Theorem 4.7) is proved by transporting probes through the differential. No definition in the chain references the quantity being defined; no 'prediction' is a renamed fit; no self-citation is load-bearing for the logical derivation. The prior-work citations provide a tool (dimPbox) whose properties are independently verifiable and do not presuppose the Point-Cross construction. The derivation is self-contained against its stated definitions and assumptions. No circularity found.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 5 invented entities

No free parameters are introduced; the theory is parameter-free. The axioms are standard mathematical results (Bouligand cones, box dimension properties, total boundedness) plus the domain assumption of Euclidean space. The invented entities are the new invariants themselves, each calibrated against known geometries with falsifiable predictions.

axioms (5)
  • standard math Bouligand tangent cone properties (sequential definition, conicity, closure invariance, C^1-diffeomorphism covariance)
    Used throughout Section 2.2 and Definition 3.1 as the foundation for effective directions. Standard result in geometric measure theory.
  • domain assumption Point-Extended Box Dimension properties (monotonicity, germ dependence, closure invariance, finite-union max-formula, bi-Lipschitz invariance)
    Recalled in Section 3.4 from the author's prior work [25]. Used as the scalar dimension engine for directional contributions. The properties are standard for upper box dimension; the pointwise localization is from the author's framework.
  • standard math Existence of maximal ε-separated subsets in bounded subsets of R^n (total boundedness)
    Used in Lemma 3.31 to extract separated sets from closure traces. Standard compactness argument.
  • standard math Bi-Lipschitz homeomorphisms preserve upper box dimension
    Used in Lemma 4.5 and the Shadowing Lemma 3.35. Standard result in fractal geometry.
  • domain assumption The ambient space is R^n with Euclidean metric (formal development restricted to finite-dimensional affine spaces)
    Stated in Section 1.4: 'the formal development is carried out in affine spaces modelled on R^n.' The general topological vector space framework from [25] is mentioned but not used.
invented entities (5)
  • Point-vector dimension (dimPvec) independent evidence
    purpose: Records maximal number of linearly independent exact local segment directions at a point
    Calibrated on geometric graphs (Proposition 2.19) and finite crosses (Example 2.4). Computed via elementary linear algebra. Falsifiable: any configuration where the rank differs from the span of incident directions would contradict Proposition 2.13.
  • Point-tangential dimension (dimPtan) independent evidence
    purpose: Records the dimension of the span of the Bouligand tangent cone at a point
    Calibrated on C^1 submanifolds (Proposition 2.50: equals manifold dimension) and level sets (Proposition 2.53). Falsifiable: if dimPtan ≠ m on some C^1 m-submanifold, the calibration fails.
  • Directional contribution θA_x(v) independent evidence
    purpose: Measures maximal point-extended box complexity detected along admissible Lipschitz probes in direction v
    Bounded in [0,1] by Lemma 3.28. Equals 1 for exact segment directions (Proposition 3.48). Closure invariance proved via Shadowing Lemma 3.35. The shadowing construction is the key falsifiable step.
  • Point-Cross Dimension (dim×) independent evidence
    purpose: Aggregates directional contributions over projectively independent effective directions
    Squeezed between dimPvec and dimPtan (Corollary 3.49). Equals k on C^1 k-submanifolds (Proposition 3.51). Equals incident-edge rank on geometric graphs (Proposition 3.56). C^1-invariant (Theorem 4.7). All calibration checks are consistent.
  • Point-Box-Cross dimension (dim⊠) independent evidence
    purpose: Combined invariant taking max of dispersive (dimPbox) and directional (dim×) components
    Defined as max{dimPbox, dim×} in Definition 3.53. Inherits properties from both components. Less deeply studied than dim× in the visible portion of the manuscript.

pith-pipeline@v1.1.0-glm · 63314 in / 3574 out tokens · 608242 ms · 2026-07-10T04:24:22.103826+00:00 · methodology

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Cite this review

Pith. "Pith review of Point-dimension theory (part II): The point-cross dimension." pith.science (2026). https://pith.science/paper/EGNUKADO

@misc{pith2026260708612,
  author       = {Pith},
  title        = {Pith review of: Point-dimension theory (part II): The point-cross dimension},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EGNUKADO}},
  note         = {Machine review of arXiv:2607.08612}
}
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read the original abstract

We introduce the Point-Cross Dimension, a new pointwise invariant designed to measure the directional organization of a set at a single point. Whereas the Point-Extended Box Dimension quantifies local dispersion and covering complexity, the Point-Cross Dimension isolates a complementary layer: the coexistence of independent effective directions through the same germ. The construction assigns weights to admissible directional probes and aggregates them over projectively independent channels, thereby turning the elementary intuition of a cross into a flexible local dimension theory. This viewpoint separates phenomena that classical isotropic dimensions often collapse. A point may have small local box dispersion while carrying several independent directional channels. Conversely, large local covering complexity need not reflect genuine directional independence. We develop the theory in three successive layers. The first is a point-vector dimension, which records exact local directions. The second is a point-tangential dimension, which replaces exact directions by Bouligand effective directions. The third is the Point-Cross Dimension, which weights these effective projective channels by the point-extended box complexity detected along admissible probes. We establish the basic structural properties of these invariants and compute the resulting Point-Cross Dimension on a range of model configurations, including finite crosses, fractal coordinate frames, oscillatory germs, self-similar curves, Sierpi\'nski-type carpets, Cantor dusts, and infinite-rank outlook examples. The final part of the paper establishes comparison principles between the directional and dispersive layers of the theory.

Figures

Figures reproduced from arXiv: 2607.08612 by Nadir Maaroufi.

Figure 1.1
Figure 1.1. Figure 1.1: Foundational cross intuition: an ordinary point on a branch carries one local direction, while the intersection [PITH_FULL_IMAGE:figures/full_fig_p005_1_1.png] view at source ↗
Figure 1.2
Figure 1.2. Figure 1.2: Disk–line transverse intersection in R 3 : the disk contributes two in-plane directions and the line contributes one transverse direction. At x× the classical max rule gives 2, while the directional reading captures 3. Let us ask the question differently: what is the pointwise dimension of a geometric n-axis frame, i.e., the union of n coordinate axes in R n? All classical dimensions (Hausdorff, box, pac… view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: Comparative local models in dimension 2: smooth point, cusp, oscillatory point with one-dimensional tangent rank, oscillatory point with two-dimensional tangent rank, corner, and double point. This local picture can then be organized globally through the tangential strata. Since dimPtan{x}A takes integer values between 0 and n, it naturally stratifies the set according to its first-order directional rich… view at source ↗

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Works this paper leans on

38 extracted references · 38 canonical work pages · 2 internal anchors

  1. [1]

    Alberti, Rank one property for derivatives of functions with bounded variation,Proceedings of the Royal Society of Edinburgh

    G. Alberti, Rank one property for derivatives of functions with bounded variation,Proceedings of the Royal Society of Edinburgh. Section A123(1993), no. 2, 239–274

    work page 1993

  2. [2]

    Alberti and A

    G. Alberti and A. Marchese, On the differentiability of Lipschitz functions with respect to measures in the Euclidean space,Geometric and Functional Analysis26(2016), no. 1, 1–66

    work page 2016

  3. [3]

    Azcan, ¸ S

    H. Azcan, ¸ S. Koçak, N. Orhun and M. Üreyen, The box-counting dimension of the sine-curve,Mathematica Slovaca 49(1999), no. 3, 367–370

    work page 1999

  4. [4]

    Bate, Structure of measures in Lipschitz differentiability spaces,Journal of the American Mathematical Society 28(2015), no

    D. Bate, Structure of measures in Lipschitz differentiability spaces,Journal of the American Mathematical Society 28(2015), no. 2, 421–482

    work page 2015

  5. [5]

    Bouligand, G. (1932). Introduction à la géométrie infinitésimale directe

    work page 1932

  6. [6]

    (2003).Fractal Geometry: Mathematical Foundations and Applications, Wiley

    Falconer, K. (2003).Fractal Geometry: Mathematical Foundations and Applications, Wiley

    work page 2003

  7. [7]

    K. J. Falconer, Seventy years of fractal projections,arXiv preprintarXiv:2602.22002, 2026

    work page arXiv 2026

  8. [8]

    K. J. Falconer, J. M. Fraser and X. Jin, Sixty years of fractal projections, inFractal Geometry and Stochastics V, Progress in Probability, vol. 70, Birkhäuser, Basel, 2015, pp. 3–25

    work page 2015

  9. [9]

    K. J. Falconer, J. M. Fraser and P. Shmerkin, Assouad dimension influences the box and packing dimensions of orthogonal projections,Journal of Fractal Geometry8(2021), no. 3, 247–259

    work page 2021

  10. [10]

    K. J. Falconer and J. D. Howroyd, Projection theorems for box and packing dimensions,Mathematical Proceedings of the Cambridge Philosophical Society119(1996), 287–295

    work page 1996

  11. [11]

    Flament, D. (1994). Hermann Ganther Grassmann: La Science de la grandeur extensive, la lineale Aus- dehnungslehre, Paris: Blanchard

    work page 1994

  12. [12]

    J. M. Fraser,Assouad Dimension and Fractal Geometry, Cambridge Tracts in Mathematics, vol. 222, Cambridge University Press, 2020

    work page 2020

  13. [13]

    Guido, D., & Isola, T. (2005). Tangential dimensions I: Metric spaces.Houston Journal of Mathematics, 31, 1023–1045

    work page 2005

  14. [14]

    Guido, D., & Isola, T. (2006). Tangential dimensions II: Measures.Houston Journal of Mathematics, 32, 423–444

    work page 2006

  15. [15]

    David Carr, Northwestern University Press, 1970

    Husserl, E.The Crisis of European Sciences and Transcendental Phenomenology, trans. David Carr, Northwestern University Press, 1970. Originally published asDie Krisis der europ¨aischen Wissenschaften und die transzendentale Ph¨anomenologie, 1936

    work page 1970

  16. [16]

    J. E. Hutchinson, Fractals and self-similarity,Indiana University Mathematics Journal30(1981), no. 5, 713–747

    work page 1981

  17. [17]

    Johnson, D. M. (1977). Prelude to dimension theory: The geometrical investigations of Bernard Bolzano. Archive for History of Exact Sciences, 17, 261-295

    work page 1977

  18. [18]

    Tangents of invariant sets

    A. Käenmäki and A. Rutar, Tangents of invariant sets,arXiv preprintarXiv:2309.11971, 2023

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  19. [19]

    Kigami,Analysis on Fractals, Cambridge Tracts in Mathematics, vol

    J. Kigami,Analysis on Fractals, Cambridge Tracts in Mathematics, vol. 143, Cambridge University Press, 2001

    work page 2001

  20. [20]

    Koike, T

    S. Koike, T. Lê Loi, L. Paunescu and M. Shiota, Directional properties of sets definable in o-minimal structures, Annales de l’Institut Fourier63(2013), no. 5, 2017–2047

    work page 2013

  21. [21]

    Koike, S., & Paunescu, L. (2009). The directional dimension of subanalytic sets is invariant under bi-Lipschitz homeomorphisms.Annales de l’Institut Fourier, 59(6), 2445–2467

    work page 2009

  22. [22]

    Korkut, D

    L. Korkut, D. Vlah, D. Žubrini´c and V . Županovi´c, Wavy spirals and their fractal connection with chirps,Mathe- matical Communications21(2016), no. 2, 251–271

    work page 2016

  23. [23]

    M. L. Lapidus and M. van Frankenhuijsen,Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings, 2nd ed., Springer Monographs in Mathematics, Springer, 2013. 128 APREPRINT- JULY10, 2026

    work page 2013

  24. [24]

    Point-Dimension Theory (Part I): The Point-Extended Box Dimension

    N. Maaroufi and E. H. Zerouali, Point-dimension theory (Part I): The point-extended box dimension, preprint arXiv:2206.06271, 2022

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  25. [25]

    Maaroufi and E

    N. Maaroufi and E. H. Zerouali, Point-Extended Box Dimension,Chaos, Solitons & Fractals166(2023), 112890

    work page 2023

  26. [26]

    Mattila,Hausdorff Dimension and Capacities of Intersections of Sets in n-Space, Acta Math

    P. Mattila,Hausdorff Dimension and Capacities of Intersections of Sets in n-Space, Acta Math. 152 (1984), 77–105

    work page 1984

  27. [27]

    Mattila,Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability, Cambridge Studies in Advanced Mathematics 44, Cambridge University Press, 1995

    P. Mattila,Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability, Cambridge Studies in Advanced Mathematics 44, Cambridge University Press, 1995

    work page 1995

  28. [28]

    P. A. P. Moran, Additive functions of intervals and Hausdorff measure,Mathematical Proceedings of the Cambridge Philosophical Society42(1946), 15–23

    work page 1946

  29. [29]

    Needleman, R

    J. Needleman, R. S. Strichartz, A. Teplyaev and P. L. Yung, Calculus on the Sierpi´nski gasket I: polynomials, exponentials and power series,Journal of Functional Analysis215(2004), no. 2, 290–340

    work page 2004

  30. [30]

    E. J. Olson, J. C. Robinson and N. Sharples, Generalised Cantor sets and the dimension of products,Mathematical Proceedings of the Cambridge Philosophical Society160(2016), no. 1, 51–75

    work page 2016

  31. [31]

    Parvate and A

    A. Parvate and A. D. Gangal, Calculus on fractal subsets of real-line I: formulation,Fractals17(2009), no. 1, 53–81

    work page 2009

  32. [32]

    Paši´c and S

    M. Paši´c and S. Tanaka, Fractal oscillations of chirp functions and applications to second-order linear differential equations,International Journal of Differential Equations, 2013, Article ID 857410, 11 pp

    work page 2013

  33. [33]

    Poincaré, H. (1903). L’espace et ses trois dimensions. Revue de métaphysique et de morale, 11(3), 281-301

    work page 1903

  34. [34]

    J. C. Robinson and N. Sharples, Strict inequality in the box-counting dimension product formulas,Real Analysis Exchange38(2013), no. 1, 95–120

    work page 2013

  35. [35]

    J. E. Sampaio, Bi-Lipschitz homeomorphic subanalytic sets have bi-Lipschitz homeomorphic tangent cones, Selecta Mathematica. New Series22(2016), no. 2, 553–559

    work page 2016

  36. [36]

    Tricot, C. (1999). Courbes et dimension fractale. Springer Science&Business Media

    work page 1999

  37. [37]

    Urysohn, P. (1925). Mémoire sur les multiplicités Cantoriennes (suite). Fundamenta Mathematicae, 8(1), 225-351

    work page 1925

  38. [38]

    C. Wei, S. Wen and Z. Wen, Remarks on dimensions of Cartesian product sets,Fractals24(2016), no. 3, 1650031. 129

    work page 2016