Pith. sign in

REVIEW 2 major objections 3 minor 28 references

Singularity of maps of several variables and a problem of Mycielski concerning prevalent homeomorphisms

T0 review · 2 major / 3 minor · reviewed 2026-08-27 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper answers Mycielski's question: the prevalent homeomorphism of the unit interval has a graph of length exactly 2.

desk verdict Correctly answers Mycielski's question and gives a coherent higher-dimensional picture; the main proofs hold up with only minor blemishes. read the letter →

arxiv 2009.14106 v3 pith:LQ4IFGV6 submitted 2020-09-29 math.CA math.GN

Richárd Balka , Márton Elekes , Viktor Kiss , Márk Poór This is my paper · ORCID
classification math.CAmath.GN MSC 28A7528C1046E1554E5257S0560B05
keywords homeomorphismHaarnullprevalentgenerictypicalsingularmapHausdorffmeasuregraphlength
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

S. Banach observed that the Baire-generic homeomorphism of the unit interval has a graph of length 2. Mycielski asked whether the measure-theoretic analogue holds, namely whether all but Haar-null many homeomorphisms, in Christensen's sense, also have graph length 2. This paper proves that the answer is yes. The proof works by showing that for a homeomorphism of [0,1] the geometric condition 'graph has length 2' is equivalent to two analytic conditions: singularity (some full-measure set is mapped to a nullset) and strong singularity (zero derivative almost everywhere), and then proving that singular homeomorphisms are prevalent. For higher dimensions, the paper shows that the Baire-generic homeomorphism of the cube has infinite d-dimensional Hausdorff measure, is singular but nowhere differentiable, while the prevalent one is singular; strong singularity is Haar ambivalent.

What carries the argument

The machinery that carries the argument is the equivalence theorem in one dimension (Theorem 3.7), which converts the geometric quantity 'length of graph' into the measure-theoretic notion of singularity; this is what lets a prevalence result about singular maps be read as an answer about graph length. The prevalence result itself is powered by Christensen's Haar-null framework and a carefully chosen witness measure: averaging the conjugates of one singular homeomorphism by the power maps $\psi_s(x)=(x_1^{s_1},\dots,x_d^{s_d})$, $s_i\in[1,2]$, gives a probability measure under which every two-sided translate of the singular set has full measure (Theorem 5.7). In the higher-dimensional generic results, the load-bearing device is the appendix's density theorem (Corollary 8.2): somewhere smooth homeomorphisms are dense in Homeo([0,1]^d), proved from stable-homeomorphism and 4-manifold results; together with a 'slides' perturbation that concentrates Jacobians and a lower-semicontinuity lemma for Hausdorff measure of graphs, this yields infinite $d$-dimensional measure generically.

What would settle it

Try to uniformly approximate the wild homeomorphism of (0,1)^4 mentioned in the introduction, which is known not to be uniformly approximable by bi-Lipschitz maps, by a somewhere smooth homeomorphism; if one such map admits no uniform approximation by any somewhere smooth homeomorphism, then Corollary 8.2 is false and Theorem 6.2 fails.

Watch

Extended reading notes

Core claim

The central discovery is that the measure-theoretic dual of Banach's category result holds: in the Polish group Homeo([0,1]) with the uniform topology, the complement of the set of homeomorphisms whose graph has length 2 is Haar null, so the prevalent homeomorphism has graph length 2 (Corollary 6.1). The route is a structural equivalence (Theorem 3.7): for f∈Homeo([0,1]), the graph has length 2 iff f is singular iff f is strongly singular, where singular means f maps some Borel set of full Lebesgue measure to a Lebesgue nullset and strongly singular means f is differentiable almost everywhere with zero derivative. The paper then proves singularity is prevalent by exhibiting an explicit witness measure: fix one singular homeomorphism $f_0$ and average over the conjugations $\psi_s\circ f_0\circ \psi_t$, where $\psi_s(x_1,\dots,x_d)=(x_1^{s_1},\dots,x_d^{s_d})$ with each $s_i\in[1,2]$; every two-sided translate of the singular set gets full measure. In higher dimensions the paper establishes a sharp contrast with the one-dimensional phenomenon: for $d\ge 2$ the Baire-generic graph has infinite $d$-dimensional Hausdorff measure (Theorem 6.2), the generic homeomorphism is singular but nowhere differentiable, the prevalent one is singular (Theorem 5.7), and the strongly singular homeomorphisms form a Haar ambivalent set (Theorem 5.1). It also classifies five candidate notions of singularity and their implications (Theorem 3.2).

Load-bearing premise

The proofs of the higher-dimensional generic results rely on the appendix's approximation theorem that somewhere smooth homeomorphisms are dense in Homeo([0,1]^d); if that density failed in dimension 4 or 5, the Baire-category conclusion about infinite Hausdorff measure would not go through.

Editorial extensions

If this is right

  • For $d\ge 2$, the graph of the Baire-generic homeomorphism of the cube has infinite $d$-dimensional Hausdorff measure, so the generic higher-dimensional graph is not a rectifiable object at the expected dimension.
  • The set of homeomorphisms whose graph has infinite $d$-dimensional Hausdorff measure is compact catcher, hence not Haar null; this leaves open but does not preclude the prevalence question for infinite measure (Question 7.3).
  • For every $d\ge 1$, the prevalent homeomorphism is singular: it maps almost all of the cube into a nullset, a genuinely measure-theoretic property.
  • For $d\ge 2$, strong singularity is Haar ambivalent: the set of homeomorphisms with zero derivative almost everywhere is neither Haar null nor co-Haar null, so no prevalence statement about strong singularity can be true in either direction.
  • The five notions of singularity form a chain (1)⇒(2)⇒(3)⇔(4)⇔(5); injectivity reverses (2)⇔(3), and differentiability almost everywhere forces all five to coincide, giving a clean toolkit for future work.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The witness measure of Theorem 5.7 is explicit enough to be simulated: sampling $s,t$ uniformly in $[1,2]^d$ and composing $\psi_s\circ f_0\circ \psi_t$ with a fixed singular $f_0$ produces homeomorphisms that should be singular almost surely; numerical estimates of graph length for $d=1$ could provide a computational check of the theorem.
  • The one-dimensional equivalence suggests that any geometric invariant that can be rewritten as a measure-theoretic nullset condition may automatically be prevalent for homeomorphisms, since singularity is prevalent; testing this on other invariants could reveal a general principle.
  • The Haar ambivalence of strong singularity in $d\ge 2$ may be a manifestation of the non-abelian group structure of Homeo([0,1]^d); comparing with the abelian case of $C([0,1]^d,\mathbb{R}^d)$ may illuminate when Haar-null arguments are usable.
  • Question 7.2 (generic Hausdorff dimension of the graph in $d=4,5$) is a natural next test: the appendix's approximation theorem is the only place where the special dimensions enter, so sharpening or refuting it would settle whether the generic dimension equals $d$ in low dimensions.
Share X Bluesky LinkedIn Reddit HN

Formalized claims in Lean

  1. Claim #1: The central discovery is that the measure-theoretic dual of Banach's category result holds: in the Polish group Homeo([0,1]) with the uniform topology, the complement of the set of homeomorphisms whose graph has length 2 is Haar null, so the prevalent homeomorphism has graph length 2 (Corollary 6.1). The route is a structural equivalence (Theorem 3.7): for f∈Homeo([0,1]), the graph has length 2 if

Signed reviews

No signed human review yet.

Request a human review

A listed scientist reviews the paper for a fee and the review publishes here regardless of verdict. See the reviewers or get listed.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. The paper studies singularity of homeomorphisms of the d-cube and settles a question of Mycielski. It proves (Corollary 6.1) that for a Christensen-prevalent f∈Homeo([0,1]) the graph has length 2. This follows from a classification theorem (Theorem 3.7) equating length-2, singularity, and strong singularity in dimension 1, and a prevalence result (Theorem 5.7) for singular homeomorphisms in all dimensions. For d≥2 the authors show that the generic homeomorphism is singular but not strongly singular, and that its graph has infinite d-dimensional Hausdorff measure, while the strongly singular homeomorphisms form a Haar ambivalent set; they also show that the set of homeomorphisms with infinite H^d(graph) is not Haar null. The Appendix establishes a density result for somewhere smooth homeomorphisms, using results of Moise, Kirby, and Freedman-Quinn.

Significance. If correct, the paper resolves Mycielski's problem in the affirmative and clarifies the higher-dimensional analogues. The main result is a genuine advance, and the proof is largely self-contained: Theorem 3.7 is a clean elementary equivalence, and the prevalence witness measure in Theorem 5.7 is explicit. The paper is careful to separate the Baire-category and Haar-null theories and to identify where external deep results enter; notably, Corollary 6.1 does not depend on the Appendix. The remaining issues are local and repairable, but they should be fixed before publication.

major comments (2)
  1. [Theorem 5.7 (proof of Borelness of F)] The proof that F is Borel asserts that the sets {(f,x): g_n(f,x)<1/k} are open. This is not justified and can fail for d≥2 because the Lebesgue measure of f(B(x,1/n)) is not continuous in f for the uniform topology when the image of the ball has a wild boundary. Since this Borelness step is needed to apply the definition of prevalence to F, the proof as written has a gap. The gap is repairable: g_n is Borel, for example because λ_d(f(B(x,1/n))) = ∫ 1_{B(x,1/n)}(f^{-1}(y)) dy and (f,y) ↦ f^{-1}(y) is Borel under a compatible complete metric on Homeo. The authors should replace 'open' with 'Borel' and supply such an argument.
  2. [Claim 6.7 / Lemma 6.5] The lower-semicontinuity statement used to obtain the dense open sets in Theorem 6.2 depends on Claim 6.7, whose proof is merely referred to [24, Theorem 10.14] 'after the straightforward modifications'. This is a load-bearing step for the generic infinite-H^d result. Please either include a proof of Claim 6.7 in the paper or state the precise theorem from [24] that applies and explain the modifications. As written, the reader cannot verify the main generic higher-dimensional result without consulting an external book and reconstructing the argument.
minor comments (3)
  1. [Section 6 (opening)] The sentence beginning 'New we turn to Question 1.5' contains a typo: 'New' should be 'Now'.
  2. [Question 7.1] The phrase 'asks weather this can be slightly improved' contains a typo: 'weather' should be 'whether'.
  3. [Lemma 4.3] The sequence q_n defined at the end of the proof is not explicitly nonincreasing near the initial segment n<a_0; this can be fixed by defining q_n=18·2^{-2m} for the largest m with a_m≤n (and a constant value, say 18, for n<a_0), so that q_n↓0 as stated.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: Corollary 6.1 follows from an independently constructed witness measure and a self-contained d=1 equivalence theorem.

full rationale

I traced the main derivation chain. Corollary 6.1 is the conjunction of Theorem 5.7 (prevalent homeomorphisms are singular) and Theorem 3.7 (for a homeomorphism of [0,1], singularity is equivalent to graph length 2). Theorem 5.7 constructs the witness measure mu from a fixed singular homeomorphism f0 using coordinate-power conjugacies psi_s and psi_t, and then proves that every left/right translate of the singular set has full mu-measure. The proof uses only Fubini's theorem, the Lipschitz properties of the maps psi_s, and the nullset-preservation lemma R_g; it does not assume the prevalence conclusion. Theorem 3.7 is proved directly by projecting the graph to the coordinate axes and by a lower-derivative argument, with no external input. The higher-dimensional results are separate: Theorem 6.2 uses Lemma 6.4, Lemma 6.5, and the Appendix approximation result, which is attributed to Luukkainen and proved from Kirby's theorem and Freedman-Quinn's Theorem 8.1A, not from the authors' own prior claims. The only self-citation I found is to the survey [10] for the standard fact that compact catcher sets are not Haar null (Fact 2.1); this is independent standard material and is not load-bearing in a circular way. None of the paper's predictions is a fitted parameter renamed as a conclusion, and no uniqueness or ansatz is imported from the authors' earlier work. The paper also explicitly leaves the prevalent higher-dimensional Hausdorff measure question open (Question 7.3), which is consistent with the absence of circular overclaiming. Overall, I find no step where a claimed derivation reduces by construction to its own input.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The central claims rely on standard background results: Christensen's Haar-null/prevalence framework, geometric measure theory facts (Rademacher, Area Formula, covering theorems), Brouwer fixed point theorem, and, in the Appendix, Kirby's stability theorem (d≥5) and Freedman-Quinn approximation (d=4). No free parameters or invented entities are introduced; all constants are part of proof constructions, not fitted to data.

assumptions (8)
  • standard math Christensen's Haar null ideal properties (proper σ-ideal, equivalence with Haar measure on locally compact groups)
    Foundation for prevalence statements in Sections 5 and 6; cited from [7,10].
  • standard math Brouwer fixed point theorem
    Used in Fact 3.4 and Claim 3.5 for continuous maps.
  • standard math Rademacher's theorem and the Area Formula
    Used in Lemma 6.4 to compute H^d(graph(g|Q)) via the Jacobian.
  • standard math 5r-covering theorem / Vitali covering theorem
    Used in Theorem 3.2, direction (2) implies (5).
  • standard math Kirby stable homeomorphism theorem (d≥5) and Freedman-Quinn approximation (d=4)
    Used in Appendix Theorem 8.1 to approximate homeomorphisms by somewhere smooth ones.
  • standard math Existence of meager full-measure sets in [0,1]^d
    Used in the proof of Theorem 4.5.
  • standard math Theorem 5.6 (measurability of sections)
    Used in Theorem 5.7 to show the singular homeomorphisms form a Borel set.
  • standard math Mass Distribution Principle
    Used in Theorem 6.3 to convert measure upper bounds into H^d lower bounds.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Singularity of maps of several variables and a problem of Mycielski concerning prevalent homeomorphisms." pith.science (2026). https://pith.science/paper/LQ4IFGV6

@misc{pith2026200914106,
  author       = {Pith},
  title        = {Pith review of: Singularity of maps of several variables and a problem of Mycielski concerning prevalent homeomorphisms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LQ4IFGV6}},
  note         = {Machine review of arXiv:2009.14106}
}
abstract

S. Banach pointed out that the graph of the generic (in the sense of Baire category) element of $\text{Homeo}([0,1])$ has length $2$. J. Mycielski asked if the measure theoretic dual holds, i.e., if the graph of all but Haar null many (in the sense of Christensen) elements of $\text{Homeo}([0,1])$ have length $2$. We answer this question in the affirmative. We call $f \in \text{Homeo}([0,1]^d)$ singular if it takes a suitable set of full measure to a nullset, and strongly singular if it is almost everywhere differentiable with singular derivative matrix. Since the graph of $f \in \text{Homeo}([0,1])$ has length $2$ iff $f$ is singular iff $f$ is strongly singular, the following results are the higher dimensional analogues of Banach's observation and our solution to Mycielski's problem. We show that for $d \ge 2$ the graph of the generic element of $\text{Homeo}([0,1]^d)$ has infinite $d$-dimensional Hausdorff measure, contrasting the above result of Banach. The measure theoretic dual remains open, but we show that the set of elements of $\text{Homeo}([0,1]^d)$ with infinite $d$-dimensional Hausdorff measure is not Haar null. We show that for $d \ge 2$ the generic element of $\text{Homeo}([0,1]^d)$ is singular but not strongly singular. We also show that for $d \ge 2$ almost every element of $\text{Homeo}([0,1]^d)$ is singular, but the set of strongly singular elements form a so called Haar ambivalent set (neither Haar null, nor co-Haar null). Finally, in order to clarify the situation, we investigate the various possible definitions of singularity for maps of several variables, and explore the connections between them.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

28 extracted references · 27 canonical work pages

  1. [24]

    Morel, S

    J. Morel, S. Solimini, Variational methods in image segmentation. With seven image process- ing experiments. Progress in Nonlinear Differential Equations and their Applications, Vol. 14, Birkh¨ auser Boston, Inc., Boston, MA, 1995

    work page 1995

  2. [1]

    Ab´ ert, T

    M. Ab´ ert, T. Keleti, Shuffle the plane,Proc. Amer. Math. Soc. 130 (2002), 549–553

    work page 2002

  3. [2]

    Balka, Z

    R. Balka, Z. Buczolich, M. Elekes, Topological Hausdorff dimension and level sets of generic continuous functions on fractals, Chaos Solitons Fractals 45 (2012), no. 12, 1579–1589

    work page 2012

  4. [3]

    Balka, U

    R. Balka, U. B. Darji, M. Elekes, Bruckner–Garg-Type results with respect to Haar null sets in C[0, 1], Proc. Edinb. Math. Soc. (2) 60 (2017), no. 1, 17–30. 30 RICH ´ARD BALKA, M ´ARTON ELEKES, VIKTOR KISS, AND M ´ARK PO ´OR

    work page 2017

  5. [4]

    Banach, ¨Uber die Baire’sche Kategorie gewisser Funktionenmengen, Studia Math

    S. Banach, ¨Uber die Baire’sche Kategorie gewisser Funktionenmengen, Studia Math. 3 (1931), 174–179

    work page 1931

  6. [5]

    A. S. Besicovitch, Parametric surfaces II. Lower semicontinuity of the area, Math. Proc. Cam- bridge Philos Soc. 45 (1949), no. 1, 14–23

    work page 1949

  7. [6]

    Burago, Y

    D. Burago, Y. Burago, S. Ivanov, A course in metric geometry , Graduate Studies in Mathe- matics, Vol. 33, American Mathematical Society, Providence, RI, 2001

    work page 2001

  8. [7]

    J. P. R. Christensen, On sets of Haar measure zero in abelian Polish groups, Israel J. Math. 13 (1972), 255–260

    work page 1972

Show all 28 references
  1. [8]

    F. R. Gantmacher, The theory of matrices , Vol. 1, AMS Chelsea Publishing, Providence, RI, 1998

    1998

  2. [9]

    S. K. Donaldson, D. P. Sullivan, Quasiconformal 4-manifolds, Acta Math. 163 (1989), 181–252

    1989

  3. [10]

    Elekes, D

    M. Elekes, D. Nagy, Haar null and Haar meager sets: a survey and new results, Bull. Lond. Math. Soc. 52, (2020) 561–619

    2020

  4. [11]

    Federer, Hausdorff measure and Lebesgue area, Proc

    H. Federer, Hausdorff measure and Lebesgue area, Proc. Nat. Acad. Sci. USA 37 (1951), 90–94

    1951

  5. [12]

    Federer, Geometric Measure Theory, Springer-Verlag, New York, 1969

    H. Federer, Geometric Measure Theory, Springer-Verlag, New York, 1969

    1969

  6. [13]

    M. H. Freedman, F. Quinn, Topology of 4-manifolds. Princeton Mathematical Series, 39. Princeton University Press, Princeton, NJ, 1990

    1990

  7. [14]

    Fulton, Algebraic Topology

    W. Fulton, Algebraic Topology. A First Course. Springer-Verlag, 1995

    1995

  8. [15]

    S. Graf, R. D. Mauldin, S. C. Williams, Random homeomorphisms, Adv. Math. 60 (1986), 239–359

    1986

  9. [16]

    B. R. Hunt, The prevalence of continuous nowhere differentiable functions,Proc. Amer. Math. Soc. 122 (1994), 711–717

    1994

  10. [17]

    A. S. Kechris, Classical descriptive set theory. Graduate Texts in Mathematics, 156, Springer- Verlag, New York, 1995

    1995

  11. [18]

    R. C. Kirby, Stable homeomorphisms and the annulus conjecture. Ann. of Math. (2) 89 (1969), 575–582

    1969

  12. [19]

    Lebesgue, Int´ egrale, longeur, aire,Ann

    H. Lebesgue, Int´ egrale, longeur, aire,Ann. Mat. Pura ed Appl. (3) 7 (1902), 231–359

    1902

  13. [20]

    Luukkainen, personal communication

    J. Luukkainen, personal communication

  14. [21]

    Mattila, Geometry of sets and measures in Euclidean spaces

    P. Mattila, Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability. Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995

    1995

  15. [22]

    M¨ orters, Y

    P. M¨ orters, Y. Peres,Brownian Motion, with an appendix by Oded Schramm and Wendelin Werner, Cambridge University Press, 2010

    2010

  16. [23]

    E. E. Moise, Geometric topology in dimensions 2 and 3. Graduate Texts in Mathematics, Vol. 47, Springer-Verlag, New York-Heidelberg, 1977

    1977

  17. [25]

    Mycielski, personal communication

    J. Mycielski, personal communication

  18. [26]

    J. C. Oxtoby, Measure and Category. A survey of the analogies between topological and measure spaces. Second edition. Graduate Texts in Mathematics No. 2, Springer-Verlag, 1980

    1980

  19. [27]

    Rudin, Real and complex analysis

    W. Rudin, Real and complex analysis. Third edition. McGraw-Hill Book Co., New York, 1987

    1987

  20. [28]

    T. B. Rushing, Topological embeddings. Pure and Applied Mathematics, Vol. 52 Academic Press, New York–London, 1973. SINGULARITY OF FUNCTIONS OF SEVERAL VARIABLES 31 Alfr´ed R´enyi Institute of Mathematics, Re´altanoda u. 13–15, H-1053 Budapest, Hun- gary Email address: balka.r...

    1973

Pith tools

Reviewed August 27, 2026 · model on record in the stance chip above.