On the graph of the dimension function of the Lagrange and Markov spectra

Carlos Gustavo Moreira; Carlos Matheus; Polina Vytnova

arxiv: 2212.11371 · v2 · submitted 2022-12-21 · 🧮 math.NT · math.DS

On the graph of the dimension function of the Lagrange and Markov spectra

Carlos Matheus , Carlos Gustavo Moreira , Polina Vytnova This is my paper

Pith reviewed 2026-05-24 09:55 UTC · model grok-4.3

classification 🧮 math.NT math.DS

keywords Lagrange spectrumMarkov spectrumHausdorff dimensionplateauxdimension functionCantor setsMarkov partitions

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The pith

The dimension function d(t) of the Lagrange and Markov spectra has twelve nontrivial plateaux, and the largest ten have lengths over 0.005.

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

The paper investigates the graph of d(t), the function that assigns to each t the Hausdorff dimension of the portion of the Lagrange and Markov spectra consisting of numbers at most t. By applying the fact that removing an element from the Markov partition of certain Cantor sets reduces their Hausdorff dimension, twelve intervals are found where d(t) is constant. Rigorous numerical computations then map the changes in d(t) between these intervals. This leads to the proof that the ten longest plateaux are exactly those longer than 0.005.

Core claim

Using the dimension drop property for dynamically defined Cantor sets, twelve nontrivial plateaux of d(t) are determined. Numerical approximations of the graph between these plateaux are produced, and as a corollary the largest ten non-trivial plateaux of d(t) are shown to be precisely those with lengths greater than 0.005.

What carries the argument

The property that the Hausdorff dimension of dynamically defined Cantor sets decreases after erasing an element of its Markov partition, applied to locate constant intervals in d(t).

If this is right

The twelve plateaux are located using the dimension drop property.
Numerical methods approximate the graph of d(t) between the plateaux.
The largest ten non-trivial plateaux are those with lengths > 0.005.

Where Pith is reading between the lines

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

The approach may help identify plateaux in related dimension functions for other Diophantine spectra.
Smaller plateaux below the 0.005 threshold remain to be classified by similar or refined methods.

Load-bearing premise

The Hausdorff dimension drop after erasing a Markov partition element applies directly to the Cantor sets defining the Lagrange and Markov spectra to determine the plateaux.

What would settle it

A precise computation revealing that d(t) is not constant on one of the twelve intervals or that there exists a plateau longer than 0.005 outside the identified ones.

read the original abstract

We study the graph of the function $d(t)$ encoding the Hausdorff dimensions of the classical Lagrange and Markov spectra with half-infinite lines of the form $(-\infty, t)$. For this sake, we use the fact that the Hausdorff dimension of dynamically Cantor sets drop after erasing an element of its Markov partition to determine twelve nontrivial plateaux of $d(t)$. Next, we employ rigorous numerical methods (from our recent joint paper with Pollicott) to produce approximations of the graph of $d(t)$ between these twelve plateaux. As a corollary, we prove that the largest ten non-trivial plateaux of $d(t)$ are exactly those plateaux with lengths $> 0.005$.

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.

Desk Editor's Note private letter to a colleague

The paper finds twelve explicit plateaux in d(t) via dimension drops and uses cited rigorous numerics to argue the ten longest are exactly those exceeding length 0.005.

read the letter

The new piece is the concrete identification of twelve nontrivial plateaux in the dimension function d(t) for the Lagrange and Markov spectra, plus the corollary that the ten largest have lengths above 0.005. The plateaux come from applying the known drop in Hausdorff dimension when an element is removed from a Markov partition; that step is straightforward and reuses established facts about these Cantor sets. The numerical approximations between the plateaux come from the authors' earlier joint work with Pollicott and are presented as rigorous enough to rule out other intervals longer than 0.005. If the error control in that prior work really gives a uniform modulus fine enough for the threshold, the corollary follows directly. The abstract does not spell out the modulus details, so the strength of the 'exactly' claim depends on how the numerics are justified in the full text. This is a narrow, technical result aimed at people already working on the spectra and their dimension functions. It does not open new directions but sharpens the picture of the graph of d(t) in a way that specialists will want to check. The theoretical part looks clean; the numerical support is the part that needs referee scrutiny for bound quality. I would send it to peer review rather than desk-reject it, because the claims are specific and the methods are standard in the subfield.

Referee Report

2 major / 1 minor

Summary. The manuscript studies the graph of the function d(t) giving the Hausdorff dimension of the portions of the classical Lagrange and Markov spectra lying in (–∞, t]. It applies the known dimension-drop property for dynamically defined Cantor sets after removal of a Markov partition element to identify twelve nontrivial plateaux of d(t). Rigorous numerical approximations (drawn from the authors’ prior joint work with Pollicott) are then used to describe the behavior of d(t) in the complementary intervals. The central corollary asserts that the ten largest nontrivial plateaux are precisely those whose lengths exceed 0.005.

Significance. If the numerical error bounds are shown to be sufficient, the result supplies a concrete and falsifiable description of the largest plateaux of d(t), combining a dynamical-systems argument with cited computational techniques. The explicit use of an established dimension-drop fact and the reference to prior rigorous numerics constitute a methodological strength.

major comments (2)

[statement of the corollary and the paragraph describing the numerical approximations] The corollary that the largest ten nontrivial plateaux are exactly those of length >0.005 rests on two load-bearing parts: the dynamical identification of twelve plateaux and the numerical verification that no additional plateaux longer than 0.005 exist in the complementary intervals. The manuscript must explicitly record the error bounds, modulus of continuity, or plateau-detection threshold furnished by the cited Pollicott joint paper that rigorously exclude undetected flat segments of length >0.005; without this, the “exactly” claim is not fully supported by the dynamical argument alone.
[section identifying the twelve plateaux] The application of the dimension-drop property after erasing a Markov partition element is invoked to locate the twelve plateaux, but the text should clarify which specific partition elements are removed and in which intervals of the spectra this produces the observed plateaux (including any dependence on the continued-fraction expansion or the Gauss map).

minor comments (1)

[introduction] Notation for the spectra and for d(t) should be introduced once with a single consistent definition rather than reappearing in slightly varying forms.

Simulated Author's Rebuttal

2 responses · 0 unresolved

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

read point-by-point responses

Referee: The corollary that the largest ten nontrivial plateaux are exactly those of length >0.005 rests on two load-bearing parts: the dynamical identification of twelve plateaux and the numerical verification that no additional plateaux longer than 0.005 exist in the complementary intervals. The manuscript must explicitly record the error bounds, modulus of continuity, or plateau-detection threshold furnished by the cited Pollicott joint paper that rigorously exclude undetected flat segments of length >0.005; without this, the “exactly” claim is not fully supported by the dynamical argument alone.

Authors: We agree that explicit reference to the error bounds is needed to fully support the claim. The joint work with Pollicott supplies rigorous error bounds together with a modulus of continuity and a plateau-detection threshold. In the revision we will quote the precise numerical thresholds and error estimates from that paper, confirming that they exclude any undetected flat segments longer than 0.005 in the complementary intervals. revision: yes
Referee: The application of the dimension-drop property after erasing a Markov partition element is invoked to locate the twelve plateaux, but the text should clarify which specific partition elements are removed and in which intervals of the spectra this produces the observed plateaux (including any dependence on the continued-fraction expansion or the Gauss map).

Authors: We will expand the relevant section to name the precise Markov partition elements removed for each plateau, specify the resulting intervals inside the Lagrange and Markov spectra, and indicate the dependence on the continued-fraction expansion and the Gauss map that produces the dimension drop in each case. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to numerical methods paper supports corollary but is not load-bearing for main dimension-drop derivation of plateaux.

full rationale

The paper's core derivation identifies twelve plateaux via the known fact that Hausdorff dimension drops after erasing a Markov partition element from dynamically defined Cantor sets; this is invoked as an established dynamical fact without reduction to self-citation or fitting. The rigorous numerical approximations (cited from the joint paper with Pollicott) are used only to bound the graph between these plateaux and support the corollary on the largest ten. This constitutes one minor self-citation that does not make the central claim equivalent to its inputs by construction. No self-definitional, fitted-input-as-prediction, or uniqueness-imported patterns appear in the derivation chain. The result is therefore self-contained against external benchmarks with only incidental self-reference.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of the dimension drop property to the spectra and the reliability of the numerical methods referenced from prior work.

axioms (1)

domain assumption Hausdorff dimension of dynamically defined Cantor sets drops after erasing an element of its Markov partition
Invoked to determine the twelve plateaux

pith-pipeline@v0.9.0 · 5648 in / 1203 out tokens · 47959 ms · 2026-05-24T09:55:06.530310+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Gauss-Cantor sets Ki defined by forbidden subsequences FW_i on {1,2} continued fractions; d(t)=min{1,2·D(t)}

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

14 extracted references · 14 canonical work pages

[1]

Bumby, Hausdorff dimensions of Cantor sets, J

R. Bumby, Hausdorff dimensions of Cantor sets, J. Reine Angew. Math. 331 (1982), 192–206

work page 1982
[2]

Cusick, The largest gaps in the lower Markoff spectrum, Duke Math

T. Cusick, The largest gaps in the lower Markoff spectrum, Duke Math. J. 41 (1974), 453–463

work page 1974
[3]

Cusick and M

T. Cusick and M. Flahive, The Markoff and Lagrange spectra , Mathematical Surveys and Monographs, 30. American Mathematical Society, Providence, RI, 1989. x+97 pp

work page 1989
[4]

Erazo, C

H. Erazo, C. G. Moreira, R. Gutiérrez-Romo, S. Romaña Fractal dimensions of the Markov and Lagrange spectra near 3. ArXiv:2208.14830

work page arXiv
[5]

G. Freiman, Diofantovy priblizheniya i geometriya chisel (zadacha Markova) [Diophantine approximation and geometry of numbers (the Markov problem)], Kalininskii Gosudarstvennyi Universitet, Kalinin, 1975

work page 1975
[6]

Hall, On the sum and product of continued fractions, Ann

M. Hall, On the sum and product of continued fractions, Ann. of Math. (2) 48 (1947), 966–993

work page 1947
[7]

Lima and C

D. Lima and C. G. Moreira, Phase transitions on the Markov and Lagrange dynamical spectra , Ann. Inst. H. Poincaré C Anal. Non Linéaire 38 (2021), 1429–1459

work page 2021
[8]

Markoff, Sur les formes quadratiques binaires indéﬁnies, Math

A. Markoff, Sur les formes quadratiques binaires indéﬁnies, Math. Ann. 15 (1879) pp. 381–406

work page
[9]

Markoff, Sur les formes quadratiques binaires indéﬁnies II, Math

A. Markoff, Sur les formes quadratiques binaires indéﬁnies II, Math. Ann. 17 (1880) pp. 379–399

work page
[10]

Matheus, C

C. Matheus, C. G. Moreira, M. Pollicott and P. Vytnova, Hausdorff dimension of Gauss–Cantor sets and two applications to classical Lagrange and Markov spectra, Adv. Math. 409 (2022), part B, Paper No. 108693, 70 pp

work page 2022
[11]

C. G. Moreira, Geometric properties of the Markov and Lagrange spectra, Ann. of Math. 188 (2018), 145–170

work page 2018
[12]

Palis and F

J. Palis and F. Takens, Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations, Fractal dimensions and inﬁnitely many attractors. Cambridge Studies in Advanced Mathematics, 35. Cambridge University Press, Cambridge, 1993. x+234 pp

work page 1993
[13]

Perron, Über die approximation irrationaler Zahlen durch rationale II, S.-B

O. Perron, Über die approximation irrationaler Zahlen durch rationale II, S.-B. Heidelberg Akad. Wiss. 8 (1921)

work page 1921
[14]

Pollicott and P

M. Pollicott and P. Vytnova, Hausdorff dimension estimates applied to Lagrange and Markov spectra, Zaremba theory, and limit sets of Fuchsian groups, preprint (2020) available at arXiv:2012.07083. To appear in Transactions of AMS. C. M ATHEUS , CNRS & É COLE POLYTECHNIQUE , CNRS (UMR 7640), 91128, P ALAISEAU , F RANCE . Email address: matheus.cmss@gmail.c...

work page arXiv 2020

[1] [1]

Bumby, Hausdorff dimensions of Cantor sets, J

R. Bumby, Hausdorff dimensions of Cantor sets, J. Reine Angew. Math. 331 (1982), 192–206

work page 1982

[2] [2]

Cusick, The largest gaps in the lower Markoff spectrum, Duke Math

T. Cusick, The largest gaps in the lower Markoff spectrum, Duke Math. J. 41 (1974), 453–463

work page 1974

[3] [3]

Cusick and M

T. Cusick and M. Flahive, The Markoff and Lagrange spectra , Mathematical Surveys and Monographs, 30. American Mathematical Society, Providence, RI, 1989. x+97 pp

work page 1989

[4] [4]

Erazo, C

H. Erazo, C. G. Moreira, R. Gutiérrez-Romo, S. Romaña Fractal dimensions of the Markov and Lagrange spectra near 3. ArXiv:2208.14830

work page arXiv

[5] [5]

G. Freiman, Diofantovy priblizheniya i geometriya chisel (zadacha Markova) [Diophantine approximation and geometry of numbers (the Markov problem)], Kalininskii Gosudarstvennyi Universitet, Kalinin, 1975

work page 1975

[6] [6]

Hall, On the sum and product of continued fractions, Ann

M. Hall, On the sum and product of continued fractions, Ann. of Math. (2) 48 (1947), 966–993

work page 1947

[7] [7]

Lima and C

D. Lima and C. G. Moreira, Phase transitions on the Markov and Lagrange dynamical spectra , Ann. Inst. H. Poincaré C Anal. Non Linéaire 38 (2021), 1429–1459

work page 2021

[8] [8]

Markoff, Sur les formes quadratiques binaires indéﬁnies, Math

A. Markoff, Sur les formes quadratiques binaires indéﬁnies, Math. Ann. 15 (1879) pp. 381–406

work page

[9] [9]

Markoff, Sur les formes quadratiques binaires indéﬁnies II, Math

A. Markoff, Sur les formes quadratiques binaires indéﬁnies II, Math. Ann. 17 (1880) pp. 379–399

work page

[10] [10]

Matheus, C

C. Matheus, C. G. Moreira, M. Pollicott and P. Vytnova, Hausdorff dimension of Gauss–Cantor sets and two applications to classical Lagrange and Markov spectra, Adv. Math. 409 (2022), part B, Paper No. 108693, 70 pp

work page 2022

[11] [11]

C. G. Moreira, Geometric properties of the Markov and Lagrange spectra, Ann. of Math. 188 (2018), 145–170

work page 2018

[12] [12]

Palis and F

J. Palis and F. Takens, Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations, Fractal dimensions and inﬁnitely many attractors. Cambridge Studies in Advanced Mathematics, 35. Cambridge University Press, Cambridge, 1993. x+234 pp

work page 1993

[13] [13]

Perron, Über die approximation irrationaler Zahlen durch rationale II, S.-B

O. Perron, Über die approximation irrationaler Zahlen durch rationale II, S.-B. Heidelberg Akad. Wiss. 8 (1921)

work page 1921

[14] [14]

Pollicott and P

M. Pollicott and P. Vytnova, Hausdorff dimension estimates applied to Lagrange and Markov spectra, Zaremba theory, and limit sets of Fuchsian groups, preprint (2020) available at arXiv:2012.07083. To appear in Transactions of AMS. C. M ATHEUS , CNRS & É COLE POLYTECHNIQUE , CNRS (UMR 7640), 91128, P ALAISEAU , F RANCE . Email address: matheus.cmss@gmail.c...

work page arXiv 2020