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arxiv: 2604.08505 · v1 · submitted 2026-04-09 · 🧮 math.PR

On d-stochastic measures with fractal support and uniform (d-1)-marginals, and related results

Nicolas Pascal Dietrich , Juan Fern\'andez S\'anchez , Wolfgang Trutschnig This is my paper

Pith reviewed 2026-05-10 16:54 UTC · model grok-4.3

classification 🧮 math.PR
keywords d-stochastic measuresfractal supportuniform marginalsHausdorff dimensionWasserstein metriciterated function systemsself-similar measuresSierpinski tetrahedron
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The pith

For d at least 3, Hausdorff dimensions of supports for measures with uniform (d-1)-marginals are dense in [d-1,d], with fractal-supported ones dense in the Wasserstein metric.

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

The paper constructs probability measures on the unit cube in d dimensions whose every (d-1)-dimensional marginal equals Lebesgue measure, using iterated function systems with probabilities. These measures can still have self-similar fractal supports. The central results show that the possible Hausdorff dimensions of such supports form a dense set inside the interval from d-1 to d, for every d at least 3. In addition, the fractal-supported measures are dense inside the full family when distance is measured by the Wasserstein metric. A concrete three-dimensional example is given whose support is a Sierpinski tetrahedron that encodes complete dependence in each coordinate direction.

Core claim

Working with iterated function systems with probabilities, we construct elements of P_d^{λ_{d-1}} that have self-similar fractal support. We prove that for every d ≥ 3 the set D_d of Hausdorff dimensions of the supports of elements in P_d^{λ_{d-1}} is dense in [d-1,d], and that the subset of elements having fractal support is dense in P_d^{λ_{d-1}} with respect to the Wasserstein metric. We also exhibit an element in P_3^{λ_2} whose support is a Sierpinski tetrahedron.

What carries the argument

Iterated function systems with probabilities (IFSPs) chosen so the invariant measure satisfies the uniform (d-1)-marginal condition while retaining a self-similar fractal support of prescribed dimension.

If this is right

  • The possible Hausdorff dimensions of supports inside the family fill [d-1,d] densely for every d ≥ 3.
  • Fractal-supported members can approximate every element of the family to any desired accuracy in Wasserstein distance.
  • There exist elements with self-similar support that realize complete functional dependence in every direction, such as the Sierpinski tetrahedron in dimension three.
  • The same IFSP construction technique extends to produce further examples of type-(ii) measures with prescribed self-similar supports.

Where Pith is reading between the lines

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

  • The constructions indicate that uniform lower-dimensional marginals do not preclude fractal geometry in the joint support, even when the dimension is arbitrarily close to the ambient space dimension.
  • Density in the Wasserstein metric suggests that fractal supports are not rare or pathological within the family but can serve as approximations to smoother measures.
  • Similar density statements might be investigated for other marginal constraints or for infinite-dimensional analogues on the unit cube.

Load-bearing premise

Suitable iterated function systems exist that produce invariant measures obeying the uniform (d-1)-marginal condition while having the claimed self-similar fractal supports and exact Hausdorff dimensions.

What would settle it

An explicit gap of positive length inside [d-1,d] containing no achievable Hausdorff dimension for any such measure, or a concrete measure in the family that cannot be approximated arbitrarily closely in Wasserstein distance by any sequence of fractal-supported ones.

Figures

Figures reproduced from arXiv: 2604.08505 by Juan Fern\'andez S\'anchez, Nicolas Pascal Dietrich, Wolfgang Trutschnig.

Figure 1
Figure 1. Figure 1: Approximation of the support of the 3-stochastic measure µ considered in Example 1; in this case the support of µ is a Sierpinski tetrahedron. 2 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Support of V n σ pλ3q as considered in Example 1 for n P t1, 2, 3, 5u; [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Density of V n τ pµλ3 q for n P t1, 2, 3u with τ according to Example 3; the color gradient going from blue (low density) via red to yellow (high density). Then we have τ P U N d , the support of τ has cardinality Nd´1 , and the induced IFSP consists of exactly Nd´1 similarities, each having shrinking factor 1 N . Loosely speaking, our idea of proof is to consider all possible permutations of τ and to work… view at source ↗
read the original abstract

The family $\mathcal{P}_{d}^{\lambda_{d-1}}$ of all probability measures on $[0,1]^d$ whose $(d-1)$-dimensional marginals are all equal to the Lebesgue measure $\lambda_{d-1}$ on $[0,1]^{d-1}$ contains remarkably pathological elements: Working with Iterated Function Systems with Probabi\-lities (IFSPs) we construct measures $\mu \in \mathcal{P}_{d}^{\lambda_{d-1}}$ of the following two types: (i) $\mu$ has self-similar fractal support; (ii) $\mu$ has self-similar support and models the situation of complete/functional dependence in each direction.As our main results concerning type (i) we prove, firstly, that for every $d\geq 3$ the set $\mathcal{D}_d$ of Hausdorff dimensions of the supports of elements in $\mathcal{P}_{d}^{\lambda_{d-1}}$ is dense in $[d-1,d]$; and, secondly, that the subset of elements in $\mathcal{P}_{d}^{\lambda_{d-1}}$ having fractal support is dense in $\mathcal{P}_{d}^{\lambda_{d-1}}$ with respect to the Wasserstein metric. Moreover, we show the existence of an element in $\mathcal{P}_{3}^{\lambda_{2}}$ of type (ii) whose support is a Sierpinski tetrahedron and study some generalizations.

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

2 major / 2 minor

Summary. The manuscript constructs elements of the family P_d^{λ_{d-1}} of probability measures on [0,1]^d with all (d-1)-marginals equal to Lebesgue measure λ_{d-1}, using iterated function systems with probabilities (IFSPs). The main results are that for d ≥ 3 the set D_d of Hausdorff dimensions of supports of self-similar fractal elements is dense in [d-1,d], and that the fractal-supported measures are dense in P_d^{λ_{d-1}} under the Wasserstein metric. It also constructs an explicit element in P_3^{λ_2} whose support is a Sierpinski tetrahedron and models complete dependence in each coordinate direction, along with some generalizations.

Significance. If the IFSP constructions are valid, the results establish that the uniform marginal constraint permits a rich variety of self-similar fractal supports whose dimensions fill [d-1,d] densely, together with density of such measures in the Wasserstein topology. This clarifies the geometric flexibility of measures with fixed marginals and connects iterated-function-system techniques to the study of singular measures with prescribed projections.

major comments (2)
  1. [Main results on type-(i) measures] The central density claims for D_d and for the Wasserstein density of fractal elements rest on the existence of IFSPs whose unique invariant measure μ satisfies π_i#μ = λ_{d-1} exactly for every coordinate projection π_i while the attractor has Hausdorff dimension taking values arbitrarily close to any point of [d-1,d]. The manuscript must supply the explicit maps and probability vectors for a sequence of such systems (at least for dimensions approaching d-1) and verify that the projected IFS reproduces Lebesgue measure without additional assumptions on the contraction ratios.
  2. [Type-(ii) construction] For the Sierpinski-tetrahedron example in P_3^{λ_2}, the paper asserts that the support models complete/functional dependence in each direction while preserving uniform marginals. The verification that the three coordinate projections of the invariant measure are exactly λ_2 (rather than merely having full support) should be written out explicitly, including the algebraic relations imposed on the contraction ratios and probabilities.
minor comments (2)
  1. [Introduction] Notation for the family P_d^{λ_{d-1}} and for the Wasserstein metric should be introduced once in the preliminaries and used consistently; the current usage mixes script and calligraphic fonts without explicit definition.
  2. [Abstract] The statement that the fractal-supported measures are dense should specify the topology on the space of measures (Wasserstein) already in the abstract for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. We address each major comment below.

read point-by-point responses

  1. Referee: [Main results on type-(i) measures] The central density claims for D_d and for the Wasserstein density of fractal elements rest on the existence of IFSPs whose unique invariant measure μ satisfies π_i#μ = λ_{d-1} exactly for every coordinate projection π_i while the attractor has Hausdorff dimension taking values arbitrarily close to any point of [d-1,d]. The manuscript must supply the explicit maps and probability vectors for a sequence of such systems (at least for dimensions approaching d-1) and verify that the projected IFS reproduces Lebesgue measure without additional assumptions on the contraction ratios.

    Authors: We appreciate the referee's emphasis on explicitness. The manuscript presents a parameterized family of IFSPs whose attractors achieve the required marginals exactly and whose dimensions are dense in [d-1,d]. To strengthen the presentation, the revised version will include an explicit sequence of maps and probability vectors (for d=3 and higher) realizing dimensions approaching d-1, together with a direct verification that each projected IFS has Lebesgue measure as its unique invariant measure. This verification proceeds from the invariance equation and the chosen positive probabilities summing to one, without further assumptions on the contraction ratios. revision: yes

  2. Referee: [Type-(ii) construction] For the Sierpinski-tetrahedron example in P_3^{λ_2}, the paper asserts that the support models complete/functional dependence in each direction while preserving uniform marginals. The verification that the three coordinate projections of the invariant measure are exactly λ_2 (rather than merely having full support) should be written out explicitly, including the algebraic relations imposed on the contraction ratios and probabilities.

    Authors: We agree that an expanded verification improves clarity. The revised manuscript will contain an explicit computation of the three coordinate projections of the invariant measure for the Sierpinski tetrahedron example. We will derive and display the algebraic relations satisfied by the contraction ratios and probabilities that force each marginal to be exactly λ_2, thereby confirming both the uniform marginals and the complete dependence property. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper constructs the required measures explicitly via iterated function systems with probabilities (IFSPs) chosen to satisfy the uniform (d-1)-marginal condition while producing self-similar supports of controlled Hausdorff dimension. The density statements for D_d in [d-1,d] and for fractal-supported elements in the Wasserstein topology follow directly from varying the contraction ratios and probabilities within the standard IFS framework and applying classical results on Hausdorff dimension and weak convergence; no equation reduces a claimed prediction back to a fitted input, no load-bearing uniqueness theorem is imported via self-citation, and no ansatz is smuggled in. The derivation chain is therefore self-contained against external benchmarks of IFS theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The results rest on standard facts about invariant measures of IFS, the formula for Hausdorff dimension of self-similar sets, and the definition of Wasserstein distance. No free parameters are introduced in the abstract; the constructions appear to use freely chosen contraction ratios and probabilities that are then shown to satisfy the marginal condition.

axioms (2)
  • standard math Invariant measures of contractive IFS exist and are unique when the maps are contractions on a complete metric space.
    Invoked implicitly when constructing the self-similar measures via IFSPs.
  • standard math Hausdorff dimension of a self-similar set equals the solution of the similarity-dimension equation under the open-set condition or equivalent separation.
    Used to compute and control the dimension of the support.

pith-pipeline@v0.9.0 · 5579 in / 1528 out tokens · 39547 ms · 2026-05-10T16:54:13.762237+00:00 · methodology

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Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

  1. [1]

    Barnsley,Fractals everywhere

    M.F. Barnsley,Fractals everywhere. Academic Press, Cambridge, 1993

    work page 1993

  2. [2]

    Dudley,Real Analysis and Probability, Cambridge University Press, 2002

    R.M. Dudley,Real Analysis and Probability, Cambridge University Press, 2002

    work page 2002

  3. [3]

    Durante, C

    F. Durante, C. Sempi,Principles of copula theory, CRC Press, Boca Raton, FL, 2016

    work page 2016

  4. [4]

    Falconer,Fractal geometry, John Wiley & Sons, Ltd, 2003

    K. Falconer,Fractal geometry, John Wiley & Sons, Ltd, 2003

    work page 2003

  5. [5]

    Feldman, Extreme doubly stochastic measures with full support,Proc

    D. Feldman, Extreme doubly stochastic measures with full support,Proc. Am. Math. Soc.114 (1992), no. 4, 919–927

    work page 1992

  6. [6]

    Fredricks, R.B

    G. Fredricks, R.B. Nelsen, J.A. Rodríguez-Lallena, Copulas with fractal supports,Insur. Math. Econ.37(2005), 42-48

    work page 2005

  7. [7]

    Griessenberger, R.R

    F. Griessenberger, R.R. Junker, W. Trutschnig, On a multivariate copula-based dependence mea- sure and its estimation,Electron. J. Stat.16(2022), 2206-2251

    work page 2022

  8. [8]

    Edgar,Measure, Topology, and Fractal Geometry, Springer Verlag, New York, 2008

    G. Edgar,Measure, Topology, and Fractal Geometry, Springer Verlag, New York, 2008

    work page 2008

  9. [9]

    Kallenberg,Foundations of modern probability, Springer Verlag, New York, 1997

    O. Kallenberg,Foundations of modern probability, Springer Verlag, New York, 1997

    work page 1997

  10. [10]

    Lindenstrauss, A remark on extreme doubly stochastic measures,Amer

    J. Lindenstrauss, A remark on extreme doubly stochastic measures,Amer. Math. Monthly(1965) 72, 379–382 17

    work page 1965

  11. [11]

    Mikusinski, M.D

    P. Mikusinski, M.D. Taylor, Some approximations ofn-copulas,Metrika72(2010), 385–414

    work page 2010

  12. [12]

    Kunze, D

    H. Kunze, D. La Torre, F. Mendivil, E. R.Vrscay,Fractal Based Methods in Analysis, Springer, New York Dordrecht Heidelberg London, 2012

    work page 2012

  13. [13]

    Losert: Counterexamples to some conjectures about doubly stochastic measures,Pacific J

    V. Losert: Counterexamples to some conjectures about doubly stochastic measures,Pacific J. Math.99no. 2 (1982), 387-397

    work page 1982

  14. [14]

    T. Mroz, S. Fuchs, W. Trutschnig, How simplifying and flexible is the simplifying assumption in pair-copula constructions - analytic answers in dimension three and a glimpse beyond,Electron. J. Stat.15(2021), 1951-1992

    work page 2021

  15. [15]

    Nelsen,An Introduction to Copulas, Springer, New York, 2006

    R.B. Nelsen,An Introduction to Copulas, Springer, New York, 2006

    work page 2006

  16. [16]

    Royden,Real Analysis(2nd Ed.), MacMillan New York, 1968

    H.L. Royden,Real Analysis(2nd Ed.), MacMillan New York, 1968

    work page 1968

  17. [17]

    Trutschnig, On a strong metric on the space of copulas and its induced dependence measure, J

    W. Trutschnig, On a strong metric on the space of copulas and its induced dependence measure, J. Math. Anal. Appl.384(2011), 690-705

    work page 2011

  18. [18]

    Trutschnig, J

    W. Trutschnig, J. Fernández Sánchez, Idempotent and multivariate copulas with fractal support, J. Stat. Plan. Infer.142(2012), 3086-3096

    work page 2012

  19. [19]

    Tsuiki, Projected images of the Sierpinski tetrahedron and other layered fractal imaginary cubes,J

    H. Tsuiki, Projected images of the Sierpinski tetrahedron and other layered fractal imaginary cubes,J. Fractal Geom.12no. 3/4 (2025), 303–339

    work page 2025

  20. [20]

    Walters,An Introduction to Ergodic Theory, Springer New York, 1982

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

    work page 1982

  21. [21]

    Weisstein,Tetrix, From MathWorld - A Wolfram Resource.https://mathworld.wolfram.com/ Tetrix.html 18

    E. Weisstein,Tetrix, From MathWorld - A Wolfram Resource.https://mathworld.wolfram.com/ Tetrix.html 18

    work page