On the packing dimension of weighted singular matrices on fractals

Anish Ghosh; Gaurav Aggarwal

arxiv: 2412.11658 · v2 · submitted 2024-12-16 · 🧮 math.NT · math.DS

On the packing dimension of weighted singular matrices on fractals

Gaurav Aggarwal , Anish Ghosh This is my paper

Pith reviewed 2026-05-23 07:10 UTC · model grok-4.3

classification 🧮 math.NT math.DS

keywords packing dimensionsingular matricesweighted singularityfractal subsetshomogeneous dynamicsunimodular latticesDiophantine approximation

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The pith

Upper bounds on packing dimension hold for weighted singular matrices even when restricted to fractal subsets.

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

The paper establishes the first upper bounds on the packing dimension of weighted singular matrices and weighted ω-singular matrices. These bounds continue to hold after intersecting the sets with fractal subsets, a result that is new even without weights for matrices and that enlarges the known fractals for row vectors. The argument proceeds by mapping the weighted singularity conditions to the property that diagonal orbits on the space of unimodular lattices p-escape on average, then bounding the packing dimension of the corresponding points in that space. A reader would care because the work links Diophantine approximation properties of matrices to the geometry of lower-dimensional sets such as fractals.

Core claim

We provide the first known upper bounds for the packing dimension of weighted singular and weighted ω-singular matrices. We also prove upper bounds for these sets when intersected with fractal subsets. The latter results, even in the unweighted setting, are already new for matrices. Further, even for row vectors, our results enlarge the class of fractals for which bounds are currently known. We use methods from homogeneous dynamics, in particular we provide upper bounds for the packing dimension of points on the space of unimodular lattices, whose orbits under diagonal flows p-escape on average.

What carries the argument

Average p-escape rate of diagonal orbits on the space of unimodular lattices, which transfers singularity conditions into packing-dimension upper bounds.

If this is right

Weighted singular matrices admit finite packing-dimension upper bounds.
The same upper bounds apply after intersection with fractal subsets.
The bounds remain valid in the unweighted case for matrices on fractals.
For row vectors the class of admissible fractals is enlarged beyond what was previously known.

Where Pith is reading between the lines

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

The dynamical translation may extend to other Diophantine conditions whose escape rates can be controlled.
Similar packing-dimension statements could be sought for Hausdorff dimension or for other flows on homogeneous spaces.
The results suggest that singular behavior remains measure-theoretically thin inside many irregular sets.

Load-bearing premise

Translating weighted singularity conditions on matrices into the average p-escape rate of diagonal orbits on unimodular lattices preserves the packing-dimension upper bound without extra assumptions on the weights or the fractal.

What would settle it

Exhibiting a fractal subset that contains a weighted singular matrix whose packing dimension exceeds the stated upper bound would falsify the claim.

read the original abstract

We provide the first known upper bounds for the packing dimension of weighted singular and weighted $\omega$-singular matrices. We also prove upper bounds for these sets when intersected with fractal subsets. The latter results, even in the unweighted setting, are already new for matrices. Further, even for row vectors, our results enlarge the class of fractals for which bounds are currently known. We use methods from homogeneous dynamics, in particular we provide upper bounds for the packing dimension of points on the space of unimodular lattices, whose orbits under diagonal flows $p$-escape on average.

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

This paper claims the first upper bounds on packing dimension for weighted singular matrices and their fractal intersections by reducing to average p-escape rates on unimodular lattices.

read the letter

The main news is that they get the first known upper bounds for the packing dimension of weighted singular and weighted ω-singular matrices, plus the same for intersections with fractal subsets. Even the unweighted matrix case on fractals is presented as new, and the results enlarge the fractals where bounds are known for row vectors as well. They do this by establishing dimension bounds for points on the space of unimodular lattices whose diagonal orbits p-escape on average, then transferring back via the usual homogeneous-dynamics correspondence between matrices and lattice orbits. That reduction looks like a direct but non-trivial extension of earlier lattice techniques to the weighted setting. The abstract states the claims cleanly and flags the dynamical step explicitly, which is helpful. No obvious circularity or free parameters appear in the outline. The soft spot is the transfer itself: the weighted singularity conditions have to map to the escape-rate sets without extra assumptions that would affect the packing dimension, and the fractal intersections need to preserve the bound under the correspondence. Without seeing the explicit escape-rate lemmas and any dimension-distortion estimates, it is impossible to check whether those steps go through rigorously. This is squarely for people working in metric number theory and homogeneous dynamics who care about Diophantine properties on fractals or singular systems. Readers already using dynamical methods for dimension bounds will get the most out of it. It deserves a serious referee because the claims are concrete, the method is standard enough to verify, and the novelty statements are testable against the literature.

Referee Report

0 major / 2 minor

Summary. The paper claims to establish the first known upper bounds on the packing dimension of the sets of weighted singular matrices and weighted ω-singular matrices, together with upper bounds on the packing dimension of these sets intersected with certain fractal subsets. The proofs proceed via homogeneous dynamics by deriving packing-dimension upper bounds for the set of points in the space of unimodular lattices whose orbits under a diagonal flow p-escape on average.

Significance. If the transfer from weighted singularity conditions to average p-escape rates preserves the claimed dimension bounds without additional restrictions, the results would enlarge the known class of fractals to which dimension bounds apply (even in the unweighted matrix setting) and would constitute the first such bounds in the weighted case. The explicit use of packing dimension for average-escape sets in homogeneous dynamics is a technical strength.

minor comments (2)

The abstract states that the fractal-intersection results are new even in the unweighted setting; a short comparison paragraph in the introduction with the best previously known bounds (e.g., for row vectors) would help readers assess the improvement.
Notation for the weight vector ω and the escape-rate parameter p should be introduced with a single consistent definition early in the preliminaries rather than re-defined in each application.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and significance assessment, which correctly identifies the novelty of the first upper bounds in the weighted case and the extension to fractal subsets (including new results even in the unweighted matrix setting). The recommendation of 'uncertain' is noted, but with no major comments provided we have no specific points requiring response or revision.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives upper bounds on packing dimension for weighted singular matrices and their fractal intersections by transferring results on average p-escape rates of diagonal orbits in the space of unimodular lattices, using standard homogeneous dynamics correspondences. No equations or steps reduce by construction to fitted inputs, self-definitions, or self-citation chains; the central transfer is presented as preserving dimension bounds without additional assumptions on weights or fractals, and the abstract indicates reliance on external machinery rather than internal renaming or ansatz smuggling. The derivation remains self-contained against the provided claims.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the dynamical reduction is presumed to rest on standard facts from homogeneous dynamics whose details are not visible.

pith-pipeline@v0.9.0 · 5618 in / 1167 out tokens · 39466 ms · 2026-05-23T07:10:18.436125+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/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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

We use methods from homogeneous dynamics, in particular we provide upper bounds for the packing dimension of points on the space of unimodular lattices, whose orbits under diagonal flows p-escape on average.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

Theorem 1.6 … dim_P(Divergent_{a,b}(x,p) ∩ K) ≤ dim_P(K) − p/(a1+b1) (min η_l w_l)

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

17 extracted references · 17 canonical work pages

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Yann Bugeaud, Yitwah Cheung, and Nicolas Chevallier, Hausdorﬀ dimension and uniform exponents in dimen- sion two , Math. Proc. Cambridge Philos. Soc. 167 (2019), no. 2, 249–284. MR3991371

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Yann Bugeaud and Michel Laurent, On exponents of homogeneous and inhomogeneous Diophantine approxima- tion, Mosc. Math. J. 5 (2005), no. 4, 747–766, 972. MR2266457

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Yitwah Cheung, Hausdorﬀ dimension of the set of singular pairs , Ann. of Math. (2) 173 (2011), no. 1, 127–167. MR2753601

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Yitwah Cheung and Nicolas Chevallier, Hausdorﬀ dimension of singular vectors , Duke Math. J. 165 (2016), no. 12, 2273–2329. MR3544282

work page 2016
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S. Chow, A. Ghosh, L. Guan, A. Marnat, and D. Simmons, Diophantine transference inequalities: weighted, inhomogeneous, and intermediate exponents , Annali della Scuola Normale Superiore di Pisa, Classe di Sc ienze XXI (2020)

work page 2020
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Tushar Das, Lior Fishman, David Simmons, and Mariusz Urb a´ nski,A variational principle in the parametric geometry of numbers , Adv. Math. 437 (2024), Paper No. 109435, 130. MR4671568

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Alex Eskin, Gregory Margulis, and Shahar Mozes, Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture , Ann. of Math. (2) 147 (1998), no. 1, 93–141. MR1609447

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Kadyrov, D

S. Kadyrov, D. Kleinbock, E. Lindenstrauss, and G. A. Ma rgulis, Singular systems of linear forms and non-escape of mass in the space of lattices , J. Anal. Math. 133 (2017), 253–277. MR3736492 ON THE PACKING DIMENSION OF WEIGHTED SINGULAR MATRICES ON FR ACTALS 29

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Osama Khalil, Singular vectors on fractals and projections of self-simil ar measures , Geom. Funct. Anal. 30 (2020), no. 2, 482–535. MR4108614

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3, Paper No

Taehyeong Kim and Jaemin Park, On a lower bound of Hausdorﬀ dimension of weighted singular v ectors, Math- ematika 70 (2024), no. 3, Paper No. e12252, 31. MR4753868

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Lingmin Liao, Ronggang Shi, Omri Solan, and Nattalie Ta mam, Hausdorﬀ dimension of weighted singular vectors in R2, J. Eur. Math. Soc. (JEMS) 22 (2020), no. 3, 833–875. MR4055990

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Johannes Schleischitz, Metric results on inhomogeneously singular vectors , 2022

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Shah and Pengyu Yang, An upper bound of the hausdorﬀ dimension of singular vectors on aﬃne subspaces, 2023

Nimish A. Shah and Pengyu Yang, An upper bound of the hausdorﬀ dimension of singular vectors on aﬃne subspaces, 2023

work page 2023
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Omri Nisan Solan, Parametric geometry of numbers with general ﬂow , 2021. Gaurav Aggarwal, School of Mathematics, Tata Institute of Fundamental Rese arch, Mumbai, India 400005 Email address : gaurav@math.tifr.res.in Anish Ghosh , School of Mathematics, Tata Institute of Fundamental Rese arch, Mumbai, India 400005 Email address : ghosh@math.tifr.res.in

work page 2021

[1] [1]

Yann Bugeaud, Yitwah Cheung, and Nicolas Chevallier, Hausdorﬀ dimension and uniform exponents in dimen- sion two , Math. Proc. Cambridge Philos. Soc. 167 (2019), no. 2, 249–284. MR3991371

work page 2019

[2] [2]

Yann Bugeaud and Michel Laurent, On exponents of homogeneous and inhomogeneous Diophantine approxima- tion, Mosc. Math. J. 5 (2005), no. 4, 747–766, 972. MR2266457

work page 2005

[3] [3]

J. W. S. Cassels, An introduction to Diophantine approximation , Cambridge Tracts in Mathematics and Math- ematical Physics, No. 45, Cambridge University Press, New Y ork, 1957. MR87708

work page 1957

[4] [4]

Yitwah Cheung, Hausdorﬀ dimension of the set of singular pairs , Ann. of Math. (2) 173 (2011), no. 1, 127–167. MR2753601

work page 2011

[5] [5]

Yitwah Cheung and Nicolas Chevallier, Hausdorﬀ dimension of singular vectors , Duke Math. J. 165 (2016), no. 12, 2273–2329. MR3544282

work page 2016

[6] [6]

S. Chow, A. Ghosh, L. Guan, A. Marnat, and D. Simmons, Diophantine transference inequalities: weighted, inhomogeneous, and intermediate exponents , Annali della Scuola Normale Superiore di Pisa, Classe di Sc ienze XXI (2020)

work page 2020

[7] [7]

Tushar Das, Lior Fishman, David Simmons, and Mariusz Urb a´ nski,A variational principle in the parametric geometry of numbers , Adv. Math. 437 (2024), Paper No. 109435, 130. MR4671568

work page 2024

[8] [8]

Alex Eskin, Gregory Margulis, and Shahar Mozes, Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture , Ann. of Math. (2) 147 (1998), no. 1, 93–141. MR1609447

work page 1998

[9] [9]

Mathemat ical foundations and applications

Kenneth Falconer, Fractal geometry, Third, John Wiley & Sons, Ltd., Chichester, 2014. Mathemat ical foundations and applications. MR3236784

work page 2014

[10] [10]

Hutchinson, Fractals and self-similarity , Indiana Univ

John E. Hutchinson, Fractals and self-similarity , Indiana Univ. Math. J. 30 (1981), no. 5, 713–747. MR625600

work page 1981

[11] [11]

Kadyrov, D

S. Kadyrov, D. Kleinbock, E. Lindenstrauss, and G. A. Ma rgulis, Singular systems of linear forms and non-escape of mass in the space of lattices , J. Anal. Math. 133 (2017), 253–277. MR3736492 ON THE PACKING DIMENSION OF WEIGHTED SINGULAR MATRICES ON FR ACTALS 29

work page 2017

[12] [12]

Osama Khalil, Singular vectors on fractals and projections of self-simil ar measures , Geom. Funct. Anal. 30 (2020), no. 2, 482–535. MR4108614

work page 2020

[13] [13]

3, Paper No

Taehyeong Kim and Jaemin Park, On a lower bound of Hausdorﬀ dimension of weighted singular v ectors, Math- ematika 70 (2024), no. 3, Paper No. e12252, 31. MR4753868

work page 2024

[14] [14]

Lingmin Liao, Ronggang Shi, Omri Solan, and Nattalie Ta mam, Hausdorﬀ dimension of weighted singular vectors in R2, J. Eur. Math. Soc. (JEMS) 22 (2020), no. 3, 833–875. MR4055990

work page 2020

[15] [15]

Johannes Schleischitz, Metric results on inhomogeneously singular vectors , 2022

work page 2022

[16] [16]

Shah and Pengyu Yang, An upper bound of the hausdorﬀ dimension of singular vectors on aﬃne subspaces, 2023

Nimish A. Shah and Pengyu Yang, An upper bound of the hausdorﬀ dimension of singular vectors on aﬃne subspaces, 2023

work page 2023

[17] [17]

Omri Nisan Solan, Parametric geometry of numbers with general ﬂow , 2021. Gaurav Aggarwal, School of Mathematics, Tata Institute of Fundamental Rese arch, Mumbai, India 400005 Email address : gaurav@math.tifr.res.in Anish Ghosh , School of Mathematics, Tata Institute of Fundamental Rese arch, Mumbai, India 400005 Email address : ghosh@math.tifr.res.in

work page 2021