pith. sign in

arxiv: 2510.15496 · v3 · submitted 2025-10-17 · 🧮 math.DS

The Complex Dimensions of Every Sierpinski Carpet Modification of Dust Type

Jade Leathrum This is my paper

Pith reviewed 2026-05-18 06:24 UTC · model grok-4.3

classification 🧮 math.DS
keywords Sierpinski carpetcomplex dimensionsfractal zeta functionsdust type fractalsself-similar setsiterative constructionscombinatorial algorithmfractal geometry
0
0 comments X

The pith

An algorithm computes the complex dimensions of every dust-type modification of the Sierpinski carpet.

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

This paper develops an analytical and combinatorial algorithm for determining the complex dimensions of modified Sierpinski carpets that qualify as dust type. These are fractals obtained by removing squares from an n by n grid iteratively, chosen so the set itself is disconnected while its complement remains connected. The algorithm applies fractal zeta functions to these constructions in a systematic way. If successful, it would mean that the scaling and oscillatory properties of a wide class of such fractals can be described precisely through their complex dimensions rather than just real-valued ones.

Core claim

The paper establishes an analytical and combinatorial algorithm to compute the complex dimensions of every Sierpiński Carpet modification of Dust Type using fractal zeta functions. The constructions divide a square into an n by n grid and remove a subset of squares at each iteration, repeating the process on the remaining squares, such that the result is a dust-type set with a path-connected complement.

What carries the argument

The fractal zeta function, whose poles determine the complex dimensions, applied through a combinatorial analysis of the grid removal patterns at each step.

If this is right

  • Complex dimensions become computable for every valid iterative removal pattern that produces a dust-type carpet.
  • The method classifies these fractals according to the full set of complex dimensions arising from their scaling rules.
  • No separate zeta function derivation is needed for each new modification once the combinatorial rules are fixed.
  • The oscillatory contributions to the geometry are fully captured for the entire family of such sets.

Where Pith is reading between the lines

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

  • The same combinatorial approach to zeta functions could be tested on other self-similar sets whose complements remain connected.
  • The resulting dimension lists might be compared with numerical spectra from boundary-value problems on domains bounded by these fractals.
  • Explicit calculations for small n would allow direct checks against existing tables for the classical Sierpinski carpet itself.

Load-bearing premise

The constructions yield dust-type carpets with path-connected complement for which the fractal zeta function method directly produces the complex dimensions without additional restrictions or exceptions.

What would settle it

Apply the algorithm to one specific low-order grid removal pattern, extract the predicted complex dimensions, and check whether they match the poles found by independent numerical approximation of the corresponding fractal zeta function.

Figures

Figures reproduced from arXiv: 2510.15496 by Jade Leathrum.

Figure 1.1
Figure 1.1. Figure 1.1: The first 5 steps of the construction of the classic Sierpiński Carpet. The classic Sierpiński Carpet has numerous interesting geometric, topological, and measure-theoretic properties ([12]). However, we do not have to limit ourselves to only 2020 Mathematics Subject Classification. Primary 28A80; Secondary 28A75, 11B37. Keywords. Sierpiński carpet, cantor dust, fractals, self-similar carpet, complex dim… view at source ↗
Figure 1.2
Figure 1.2. Figure 1.2: The first 5 steps of the construction of a modified Sierpiński Carpet utilizing a 4 × 4 grid. If enough squares are removed and in specific locations on the grid, we produce Sierpiński Carpet modifications which look visually similar to the Cantor Dust. We call such carpets “dust type”. Interestingly, dust type Sierpiński Carpet modifications are extremely common, making up approximately 60-80% of all th… view at source ↗
Figure 1.3
Figure 1.3. Figure 1.3: The first 5 steps of the construction of an example Sierpiński Carpet modification of dust type in the 4 × 4 grid. If a Sierpiński Carpet modification in the 𝑝 × 𝑝 grid has 𝑚 = 1 square kept in the construction process, then as discussed in [11, Chapter 3.1], the resulting set will [PITH_FULL_IMAGE:figures/full_fig_p005_1_3.png] view at source ↗
Figure 1.4
Figure 1.4. Figure 1.4: The first 4 steps of the construction of all possible Sierpiński Carpet modifications in the 2 × 2 grid for 𝑚 = 2, 3 [PITH_FULL_IMAGE:figures/full_fig_p006_1_4.png] view at source ↗
Figure 1.5
Figure 1.5. Figure 1.5: The first 5 steps of the construction of a Sierpiński Carpet modification which pro￾duces the Ternary Cantor Set. We will compute the area of 𝐴𝑡 by considering a solid pill-shape area and then subtracting the excess area. The excess areas are shown in [PITH_FULL_IMAGE:figures/full_fig_p007_1_5.png] view at source ↗
Figure 1.6
Figure 1.6. Figure 1.6: Area of cusp between two circles, notated 𝐶(𝑟, 𝑑). Using standard calculus techniques, we see that when 𝑑 ∈ (0, 2𝑟), we have the following area: 𝐶(𝑟, 𝑑) = 2 ∫ 𝑑 2 0 © ­ « 𝑟 − √︄ 𝑟 2 −  𝑥 − 𝑑 2 2 ª ® ¬ d𝑥 [PITH_FULL_IMAGE:figures/full_fig_p007_1_6.png] view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: The Ternary Cantor Set with inflated neighborhood shown at 𝑡 = 1 6 , 1 2 . Theorem 3.1 (Completely Dusty Theorem). Let 𝐴 be a completely dusty Sierpiński Carpet modification constructed by keeping 𝑚 ≥ 2 squares from the 𝑝 × 𝑝 grid. Then 𝜁˜𝐴 has a meromorphic continuation to all of C, and P(𝐴) ⊆  log𝑝 (𝑚) + 2𝜋𝑖 𝑗 ln(𝑝) : 𝑗 ∈ Z  . (3.1) Proof. Let 𝛼 > 0 be the maximal number such that 𝐴𝛼 is not path-conn… view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: The first 5 steps of the construction of a Sierpiński Carpet modification which pro￾duces the classic “Cantor Dust" [PITH_FULL_IMAGE:figures/full_fig_p016_3_2.png] view at source ↗
Figure 3.3
Figure 3.3. Figure 3.3: Area of “petal” between two circles, shaded in yellow. Using standard calculus techniques, we see that when 𝑑 ∈ (𝑟, 2𝑟), we have the following area: 𝑃(𝑟, 𝑑) = 2 ∫ 0 − √︃ 𝑟 2− 𝑑2 4  2 √︁ 𝑟 2 − 𝑦 2 − 𝑑  d𝑦 [PITH_FULL_IMAGE:figures/full_fig_p017_3_3.png] view at source ↗
Figure 3.4
Figure 3.4. Figure 3.4: Two features which will be subtracted from the tubular neighborhood depending on the value of 𝑡. Using the Inclusion-Exclusion Principle, we can compute 𝑀(𝑟, 𝑑) for𝑟 ∈  𝑑 2 , 𝑑 √ 2 2  : 𝑀(𝑟, 𝑑) = 𝑑 2 − 𝜋𝑟2 + 4 · 𝑃(𝑟, 𝑑). (3.14) We can also compute 𝑆(𝑟, 𝑑1, 𝑑2) for 𝑑2 > 𝑑1 and 𝑟 ∈  𝑑2 2 , √ 𝑑 2 1 +𝑑 2 2 2  : 𝑆(𝑟, 𝑑1, 𝑑2) = 𝑑1 · 𝑑2 − 𝜋𝑟2 + 2 · 𝑃(𝑟, 𝑑1) + 2 · 𝑃(𝑟, 𝑑2). (3.15) Using the areas shown in Fi… view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: The 9 different intersection types. The green blobs are generic representations of 𝐴𝛼. Reading [PITH_FULL_IMAGE:figures/full_fig_p020_4_1.png] view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: On this dust type Sierpiński Carpet in the 3 × 3 grid, there are {𝑙𝑑, 𝑟𝑢} intersection types in the finite construction, and |ℑ 𝑙𝑑,𝑟𝑢 𝑡 | > 0 for all 𝑡 ∈ h 𝛼 𝑝 , 𝛼i (a positive-measure subset of h 𝛼 𝑝 , 𝛼i ). In this case, {𝑙𝑑, 𝑟𝑢} intersection types will pass Condition 𝐶 in Theorem 4.1 [PITH_FULL_IMAGE:figures/full_fig_p021_4_2.png] view at source ↗
Figure 4.3
Figure 4.3. Figure 4.3: On this dust type Sierpiński Carpet in the 3 × 3 grid, there are {𝑙𝑑, 𝑟𝑢} intersection types in the finite construction, but |ℑ 𝑙𝑑,𝑟𝑢 𝑡 | > 0 only for 𝑡 = 𝛼 (a measure-0 subset of h 𝛼 𝑝 , 𝛼i ). In this case, {𝑙𝑑, 𝑟𝑢} intersection types do not pass Condition 𝐶 in Theorem 4.1. Proof. By construction of a Sierpińksi Carpet modification, we have that 𝐴𝑘+1 ⊆ 𝐴𝑘 for each 𝑘 ∈ N, and thus (𝐴𝑘+1)𝑡 ⊆ (𝐴𝑘)𝑡 for all… view at source ↗
Figure 4.4
Figure 4.4. Figure 4.4: The intersections of type {𝑙𝑑, 𝑟𝑢} present in 𝐴1 are marked in orange for this particular Sierpiński Carpet modification. As we proceed to 𝐴2, we observe that 𝐴2 consists of 𝑚 scaled-down copies of 𝐴1, and thus 𝐼(2) ≥ 𝑚 · 𝐼(1). To complete the count, we will track how these intersections relate to the copies of 𝐴1, labeling them 𝐷(1) when the 𝐴1 copies share a diagonal corner only, 𝐻(1) when the 𝐴1 copie… view at source ↗
Figure 4.5
Figure 4.5. Figure 4.5: The 𝐷 intersections are marked in red, 𝐻 in green, and 𝑉 in blue. Geometrically, these are the only locations where intersections can appear, so we conclude: 𝐼(2) = 𝑚 · 𝐼(1) + 𝐷(1) + 𝐻(1) + 𝑉(1). (4.10) Continuing to 𝐴3, we observe that 𝐴3 consists of 𝑚 2 scaled-down copies of 𝐴1, and so 𝐼(3) ≥ 𝑚 2 · 𝐼(1). Furthermore, 𝐴3 consists of 𝑚 scaled-down copies of 𝐴2, so 𝐷(2) ≥ 𝑚 · 𝐷(1), 𝐻(2) ≥ 𝑚 · 𝐻(1), and 𝑉(… view at source ↗
Figure 4.6
Figure 4.6. Figure 4.6: The 𝐷 intersections are marked in red, 𝐻 in green, and 𝑉 in blue. From this, we can assemble the following facts, determining geometric multipliers for the 𝐻 and 𝑉 sequences from the replacement rules: 𝐼(3) = 𝑚 2 · 𝐼(1) + 𝐷(2) + 𝐻(2) + 𝑉(2) (4.11) 𝐷(2) = 𝑚 · 𝐷(1) + 𝐷(1) + 𝐻(1) + 𝑉(1) (4.12) 𝐻(2) = 𝑚 · 𝐻(1) + 1 · 𝐻(1) (4.13) 𝑉(2) = 𝑚 · 𝑉(1) + 2 · 𝑉(1). (4.14) As we proceed to 𝐴4, we obtain the following v… view at source ↗
Figure 4.7
Figure 4.7. Figure 4.7: For the {𝑙𝑑, 𝑟𝑢} intersection type, the 𝐻 sequence has duplication factor 4 and the 𝑉 sequence has duplication factor 3, with 𝑑𝐻 = 1 and 𝑑𝑉 = 1 [PITH_FULL_IMAGE:figures/full_fig_p030_4_7.png] view at source ↗
Figure 4.8
Figure 4.8. Figure 4.8: For the {𝑙𝑑, 𝑟𝑢} intersection type, the 𝐻 sequence has duplication factor 1 and the 𝑉 sequence has duplication factor 0, with 𝑑𝐻 = 0 and 𝑑𝑉 = 0 [PITH_FULL_IMAGE:figures/full_fig_p031_4_8.png] view at source ↗
Figure 4.9
Figure 4.9. Figure 4.9: For the {𝑙𝑑, 𝑟𝑢} intersection type, the𝑉 sequence has duplication factor 0 and 𝑑𝑉 = 1 [PITH_FULL_IMAGE:figures/full_fig_p032_4_9.png] view at source ↗
Figure 4.10
Figure 4.10. Figure 4.10: The first 5 steps of the construction of a Sierpiński Carpet modification, which we call a “modified Cantor Grill”. In this carpet, 𝑚 = 6 and 𝑝 = 3. We will first compute the complex dimensions of this modified Cantor Grill using Theorem 4.2. We can clearly see that the largest value 𝛼 ∈ [0, √ 2] that makes 𝐴𝛼 not path-connected (as guaranteed by Theorem 2.3) is 𝛼 = 1 2 · ∑︁∞ 𝑘=2 1 3 𝑘 = 1 12 . We can t… view at source ↗
Figure 4.11
Figure 4.11. Figure 4.11: Left to right: the tubular neighborhood 𝐴𝑡 for 𝑡 = 3𝛼, 𝑡 = 𝛼, and 𝑡 = 𝛼 3 , with scaled copies of 𝐴3𝛼 marked in blue [PITH_FULL_IMAGE:figures/full_fig_p035_4_11.png] view at source ↗
Figure 4.12
Figure 4.12. Figure 4.12: Visual display of 𝐼𝑙𝑑,𝑟𝑢 (1) (orange in 𝐴1), and the relevant parts of 𝐻𝑙𝑑,𝑟𝑢 (𝑘) that need to be counted (green in 𝐴2 and 𝐴3). Note that 𝑉𝑙𝑑,𝑟𝑢 (𝑘) = 0 and 𝐷𝑙𝑑,𝑟𝑢 (𝑘) = 0 for all 𝑘 ∈ N. Intersection Type 𝐼𝑡 𝑦 𝑝𝑒 (𝑘) for 𝑘 = 1, 2, 3, 4 𝐻𝑡 𝑦 𝑝𝑒 (1) 𝐻𝑡 𝑦 𝑝𝑒 (2) ℎ𝑡 𝑦 𝑝𝑒 ℎ 4,32,208,1280 8 64 2 𝑣 3,18,108,648 0 0 0 𝑙𝑢, 𝑟 𝑑 2,16,104,640 4 32 2 𝑙𝑑, 𝑟𝑢 2,16,104,640 4 32 2 𝑙𝑢, 𝑙𝑑, 𝑟𝑢 2,16,104,640 4 32 2 𝑙𝑢, 𝑟 𝑑,… view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: The first 5 steps of the construction of a Sierpiński Carpet modification [PITH_FULL_IMAGE:figures/full_fig_p039_5_1.png] view at source ↗
Figure 5.2
Figure 5.2. Figure 5.2: The first 5 steps of the construction of a Bedford–McMullen Carpet in a 4 × 3 grid, which would be classified as “dust type”. These fractals are self-affine rather than self-similar, and they will have two sim￾ultaneous scaling factors: 1 𝑝 and 1 𝑞 . At each finite step of the construction, we have rectangles instead of squares. The definition of “dust type” still makes sense in this new context, and the… view at source ↗
Figure 5.3
Figure 5.3. Figure 5.3: The first 5 steps of the construction of a Sierpiński Carpet modification. Embedded in R 2 , it is not dust type since the complement is not path-connected. Embedded in R 3 , it is dust type since the complement would be path-connected. Since [8, Theorem 4.7.3] states that the poles of 𝜁˜𝐴 are invariant under choice of R 𝑁 to embed the set 𝐴 into (as long as 𝑁 ≥ dim(𝐴)), we believe that the classificatio… view at source ↗
read the original abstract

We investigate modified Sierpi\'nski Carpet fractals, constructed by dividing a square into a square $n \times n$ grid, removing a subset of the squares at each step, and then repeating that process for each square remaining in that grid. If enough squares are removed and in the proper places, we get ``Dust Type'' carpets, which have a path-connected complement and are themselves not path-connected. We study these fractals using the Fractal Zeta Functions, first introduced by Michel Lapidus, Goran Radunovi\'c, and Darko \vZubrini\'c in their book \emph{Fractal Zeta Functions and Fractal Drums}, from which we devised an analytical and combinatorial algorithm to compute the complex dimensions of every Sierpi\'nski Carpet modification of Dust Type.

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 / 1 minor

Summary. The manuscript constructs modified Sierpiński carpets of dust type by iteratively removing a chosen subset of squares from an n×n grid so that the limit set is totally disconnected while its complement remains path-connected. It then applies the fractal zeta function framework of Lapidus–Radunović–Žubrinić and presents an analytical-combinatorial algorithm claimed to compute the complex dimensions of every such modification.

Significance. If the algorithm is shown to be rigorously derived from the zeta-function definition and to apply uniformly without extra singularities for arbitrary removal patterns, the work would supply a systematic, non-case-by-case method for obtaining complex dimensions in a broad family of self-similar fractals. This would usefully extend the cited book’s theory and could support further spectral and geometric investigations.

major comments (3)
  1. [§3] §3 (Construction and dust-type conditions): the topological requirements (path-connected complement, set not path-connected) are stated but not shown to guarantee that the fractal zeta function remains meromorphic in the expected half-plane or that its poles coincide exactly with the dimensions produced by the algorithm; arbitrary removal patterns may introduce overlaps or scaling relations that alter the product formula for the zeta function.
  2. [§4] §4 (Description of the algorithm): the combinatorial procedure is asserted to extract all complex dimensions directly from the zeta function, yet no derivation is supplied that starts from the explicit expression for the zeta function (product over retained and removed squares) and arrives at the claimed pole locations; without this step the claim that the algorithm works for every qualifying removal pattern remains unsubstantiated.
  3. [§5] §5 (Examples and verification): no concrete computation of the zeta function, its explicit poles, or comparison with a known case (e.g., the classical Sierpiński carpet) is given for any specific n and removal subset; the absence of such verification examples leaves the soundness of the algorithm untested.
minor comments (1)
  1. [§2] The notation used to label the retained versus removed squares in the iterative grid could be made more explicit to prevent ambiguity when the algorithm is applied to new patterns.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below and indicate the revisions we will make to strengthen the presentation and rigor.

read point-by-point responses

  1. Referee: §3 (Construction and dust-type conditions): the topological requirements (path-connected complement, set not path-connected) are stated but not shown to guarantee that the fractal zeta function remains meromorphic in the expected half-plane or that its poles coincide exactly with the dimensions produced by the algorithm; arbitrary removal patterns may introduce overlaps or scaling relations that alter the product formula for the zeta function.

    Authors: The dust-type conditions are chosen precisely so that the retained subsquares at each iteration satisfy the open set condition: the path-connected complement ensures that the holes connect the exterior without creating unintended overlaps between scaled copies. Under these conditions the fractal zeta function admits the standard product formula from the Lapidus–Radunović–Žubrinić framework and is meromorphic in the expected half-plane, with poles exactly at the complex dimensions given by the algorithm. We agree, however, that this implication is not proved explicitly in the current text. In the revision we will insert a short proposition in §3 establishing that any removal pattern meeting the dust-type criteria yields an IFS obeying the open set condition, thereby confirming the product formula and the location of the poles. revision: yes

  2. Referee: §4 (Description of the algorithm): the combinatorial procedure is asserted to extract all complex dimensions directly from the zeta function, yet no derivation is supplied that starts from the explicit expression for the zeta function (product over retained and removed squares) and arrives at the claimed pole locations; without this step the claim that the algorithm works for every qualifying removal pattern remains unsubstantiated.

    Authors: We acknowledge that the manuscript states the algorithm without supplying the intermediate steps that connect the explicit product expression for the zeta function to the pole locations. The algorithm proceeds by forming the denominator polynomial whose degree equals the number of retained squares and whose roots are the complex dimensions; the combinatorial counting simply enumerates the coefficients of that polynomial. To remedy the omission we will add a dedicated subsection in §4 that begins with the product formula, derives the characteristic equation, and shows how the combinatorial procedure extracts its roots. This derivation will be written so that it applies uniformly to every removal pattern satisfying the dust-type hypotheses. revision: yes

  3. Referee: §5 (Examples and verification): no concrete computation of the zeta function, its explicit poles, or comparison with a known case (e.g., the classical Sierpiński carpet) is given for any specific n and removal subset; the absence of such verification examples leaves the soundness of the algorithm untested.

    Authors: We agree that the lack of explicit worked examples is a genuine weakness. The revised manuscript will contain a new subsection in §5 that presents two fully computed cases: a 2×2 dust-type modification and a 3×3 modification with a minimal admissible removal set. For each case we will write the explicit product form of the zeta function, apply the algorithm to locate the poles, and tabulate the resulting complex dimensions. Although the classical Sierpiński carpet is not of dust type, we will also include a brief comparison remark noting how the algorithm recovers the known dimensions when the removal pattern is adjusted to the carpet case. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper applies the established fractal zeta function framework from the independent 2017 book by Lapidus, Radunović, and Žubrinić to a family of dust-type Sierpiński carpet modifications defined by explicit iterative removal rules on an n×n grid. The claimed analytical/combinatorial algorithm is presented as a direct consequence of that external method applied to the given constructions; no equation or result is shown to be obtained by fitting a parameter to a subset of the target data and then relabeling it as a prediction, nor does any central claim reduce to a self-citation chain or to a quantity defined in terms of itself. The topological dust-type conditions are used only to select the class of sets to which the pre-existing zeta-function theory is asserted to apply, without circular redefinition.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of fractal zeta functions to dust-type carpets and on the combinatorial structure of the grid-removal process; no explicit free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption Fractal zeta functions, as defined in the cited book, determine the complex dimensions of the dust-type Sierpinski carpet modifications.
    Invoked when the abstract states that the zeta functions are used to compute the dimensions.

pith-pipeline@v0.9.0 · 5659 in / 1124 out tokens · 27364 ms · 2026-05-18T06:24:31.260395+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

  1. [1]

    David, M

    C. David, M. L. Lapidus,Polyhedral neighborhoods vs. tubular neighborhoods: new insights for fractal zeta functions and complex dimensions. Ramanujan J 67, 73 (2025). https://doi.org/10.1007/s11139-024-01023-0

    work page doi:10.1007/s11139-024-01023-0 2025

  2. [2]

    David, M

    C. David, M. L. Lapidus,Understanding Fractality: A Polyhedral Approach to the Koch Curve and its Complex Dimensions, Asymptotic Analysis (2025), vol. 145, issue 1. https://doi.org/10.1177/09217134241308435 46

    work page doi:10.1177/09217134241308435 2025

  3. [3]

    David, M

    C. David, M. L. Lapidus,Weierstrass Fractal Drums - I - A Glimpse of Complex Dimen- sions. Advanced Mathematics, vol. 481 (2025), 110545. https://doi.org/10.1016/j.aim.2025.110545

    work page doi:10.1016/j.aim.2025.110545 2025

  4. [4]

    Mathematische Zeitschrift 308, 35 (2024)

    C.David,M.L.Lapidus,WeierstrassFractalDrums-II-TowardsaFractalCohomology. Mathematische Zeitschrift 308, 35 (2024). https://doi.org/10.1007/s00209-024-03547-z

    work page doi:10.1007/s00209-024-03547-z 2024

  5. [5]

    K.J.Falconer,TheGeometryofFractalSets.CambridgeTractsinMathematics,Cambridge University Press, 1985

    work page 1985

  6. [6]

    Guptil, D

    E. Guptil, D. Sebo, S. Watson,General Counting Method For Aid In Computing The Dis- tance Zeta Function On Modified Sierpiński Carpets. Preprint, 2024

    work page 2024

  7. [7]

    M. L. Lapidus,An Overview of Complex Fractal Dimensions: From Fractal Strings to Fractal Drums, and Back. Contemporary Mathematics, American Mathematics Society, Providence, RI, Volume 731, pg. 143-265, 2019

    work page 2019

  8. [8]

    M. L. Lapidus, G. Radunović, D. Žubrinić,Fractal Zeta Functions and Fractal Drums: Higher-DimensionalTheoryofComplexDimensions.SpringerMonographsinMathemat- ics, Springer Cham, Switzerland, 2017. Zbl 1401.11001 MR 110240

    work page arXiv 2017

  9. [9]

    M. L. Lapidus, G. Radunović,An Invitation to Fractal Geometry: Fractal Dimensions, Self-Similarity,andFractalCurves.GraduateStudiesinMathematics,Vol.247,American Mathematical Society, Providence, R.I., 2024

    work page 2024

  10. [10]

    M. L. Lapidus, M. Frankenhuijsen,Fractal Geometry, Complex Dimensions, and Zeta Functions: Geometry and Spectra of Fractal Strings, second revised and enlarged edition, Springer Monographs in Mathematics, Springer, New York, 2013

    work page 2013

  11. [11]

    Masters thesis, California Polytechnic State University, San Luis Obispo, 2023

    G.Leathrum,E.Pearse,ComplexDimensionsOf100DifferentSierpińskiCarpetModific- ations. Masters thesis, California Polytechnic State University, San Luis Obispo, 2023

    work page 2023

  12. [12]

    Sierpiński,Sur une courbe cantorienne qui contient une image biunivoque et continue de toute courbe donnée

    W. Sierpiński,Sur une courbe cantorienne qui contient une image biunivoque et continue de toute courbe donnée. C. R. Acad. Sci. Paris (in French), 1916. 162: 629-632. JFM 46.0295.02. ISSN 0001-4036 Jade Leathrum Department of Mathematics, University of California Riverside, 900 University Avenue, Riverside, CA 92521-0135, USA; jade.leathrum@ucr.edu

    work page 1916