Point-to-set Principle and Constructive Dimension Faithfulness

Akhil S; Satyadev Nandakumar; Subin Pulari

arxiv: 2403.08278 · v3 · submitted 2024-03-13 · 💻 cs.IT · math.IT

Point-to-set Principle and Constructive Dimension Faithfulness

Satyadev Nandakumar , Subin Pulari , Akhil S This is my paper

Pith reviewed 2026-05-24 02:59 UTC · model grok-4.3

classification 💻 cs.IT math.IT

keywords Hausdorff dimensionconstructive dimensionCantor coveringspoint-to-set principleKolmogorov complexityfaithfulnessCantor series expansion

0 comments

The pith

Cantor coverings are faithful to Hausdorff dimension exactly when faithful to constructive dimension.

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

The paper introduces constructive Φ-dimension, an effective version of Hausdorff Φ-dimension based on restricted coverings. It establishes a point-to-set principle for Φ-dimension that yields Kolmogorov complexity characterizations and applies to Hausdorff dimension, continued fraction dimension, and Cantor coverings as special cases. For coverings from the Cantor series expansion, necessary and sufficient conditions for constructive faithfulness are derived from the terms of the expansion. A new construction of sequences meeting a Kolmogorov complexity bound is used to prove that faithfulness at the Hausdorff level is equivalent to faithfulness at the constructive level. This equivalence supplies an information-theoretic derivation of the necessary and sufficient conditions for Hausdorff dimension faithfulness of Cantor coverings.

Core claim

We prove a point-to-set principle for Φ-dimension by introducing constructive Φ-dimension and characterizing it via Kolmogorov complexity. For Cantor series coverings we obtain necessary and sufficient conditions for constructive dimension faithfulness based on the expansion terms. Using the point-to-set principle together with a technique for building sequences that satisfy a prescribed Kolmogorov complexity condition, we show that the faithfulness notions coincide at the Hausdorff and constructive levels, and therefore the same conditions characterize Hausdorff dimension faithfulness.

What carries the argument

constructive Φ-dimension together with its Kolmogorov complexity characterization, which carries the point-to-set principle and enables the equivalence proof for Cantor covering faithfulness

If this is right

Point-to-set principles hold for Hausdorff dimension, continued fraction dimension, and dimension of Cantor coverings.
Necessary and sufficient conditions for constructive dimension faithfulness of Cantor coverings are stated directly in terms of the expansion coefficients.
The same conditions characterize Hausdorff dimension faithfulness of those coverings.
An information-theoretic proof is obtained for the earlier result of Albeverio, Ivanenko, Lebid and Torbin on Hausdorff faithfulness.

Where Pith is reading between the lines

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

The Kolmogorov complexity construction technique may transfer to equivalence proofs for other effective and classical dimensions.
The framework suggests that similar point-to-set arguments could link algorithmic randomness to other restricted covering classes beyond Cantor series.
Results on faithfulness may inform computable approximations of Hausdorff dimension in settings where direct calculation is intractable.

Load-bearing premise

The point-to-set principle for Φ-dimension holds via the introduced constructive Φ-dimension and its Kolmogorov complexity characterization.

What would settle it

A concrete Cantor covering for which the constructive faithfulness conditions hold but the classical Hausdorff dimension of some set differs from its Φ-dimension, or vice versa.

read the original abstract

Hausdorff $\Phi$-dimension is a notion of Hausdorff dimension developed using a restricted class of coverings of a set. We introduce an effective version of Hausdorff $\Phi$-dimension, which we call constructive $\Phi$-dimension. We prove a point-to-set principle for $\Phi$-dimension, through which we get point-to-set principles for Hausdorff dimension, continued fraction dimension, and dimension of Cantor coverings as special cases. We also provide a Kolmogorov complexity characterization of constructive $\Phi$-dimension. A class of covering sets $\Phi$ is said to be ``faithful'' to Hausdorff dimension if the $\Phi$-dimension and Hausdorff dimension coincide for every set. Similarly, $\Phi$ is said to be ``faithful'' to constructive dimension if the constructive $\Phi$-dimension and constructive dimension coincide for every set. We derive the necessary and sufficient conditions for the constructive dimension faithfulness of the coverings generated by the Cantor series expansion, based on the terms of the expansion. Using the point-to-set principle for Cantor coverings and a new technique for the construction of sequences satisfying a certain Kolmogorov complexity condition, we show that the notions of ``faithfulness'' of Cantor coverings at the Hausdorff and constructive levels are equivalent. Hence we show the necessary and sufficient conditions for Hausdorff dimension faithfulness of Cantor coverings, thereby giving an information theoretic proof of the result by Albeverio, Ivanenko, Lebid, and Torbin.

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 defines constructive Φ-dimension, proves its point-to-set principle with a Kolmogorov characterization, and shows faithfulness of Cantor coverings is equivalent at Hausdorff and constructive levels.

read the letter

The main thing to know is that this paper defines constructive Φ-dimension, proves a point-to-set principle for it that recovers Hausdorff dimension, continued fraction dimension, and Cantor covering dimension as special cases, and then shows that the faithfulness condition for Cantor coverings is the same whether you work at the Hausdorff or constructive level. That equivalence also gives an information-theoretic proof of the necessary and sufficient conditions from Albeverio et al. on Cantor series expansions. They add a technique for building sequences that meet specific Kolmogorov complexity conditions to make the equivalence go through. What is new is the constructive Φ-dimension itself, its complexity characterization, and the demonstration that the two faithfulness notions coincide. The paper does a clean job of setting up the general point-to-set tool and showing how the Cantor case specializes without obvious logical gaps at the level of the claims. The soft spot is the Kolmogorov characterization step for the new dimension. That is the load-bearing piece that lets the point-to-set principle go through and transfers the faithfulness result. The abstract says the proof is there, but any looseness in the effective covering bounds or the complexity estimates would affect the equivalence, so that section needs checking. No circularity or other red flags show up. This is for people already working in algorithmic information theory and effective fractal geometry. A reader who knows point-to-set principles or Cantor expansions would get the most out of the new conditions and the bridging technique. It deserves a serious referee because the results are new, the approach is direct, and the claims are internally consistent on the evidence given.

Referee Report

0 major / 3 minor

Summary. The manuscript introduces constructive Φ-dimension as an effective version of Hausdorff Φ-dimension. It proves a point-to-set principle for Φ-dimension (yielding corollaries for Hausdorff dimension, continued fraction dimension, and Cantor covering dimension), provides a Kolmogorov complexity characterization of constructive Φ-dimension, and for Cantor coverings shows that faithfulness at the Hausdorff and constructive levels are equivalent via a new technique for constructing sequences satisfying Kolmogorov complexity conditions. This equivalence yields necessary and sufficient conditions for Hausdorff dimension faithfulness of Cantor coverings, supplying an information-theoretic proof of a result by Albeverio et al.

Significance. If the central derivations hold, the work unifies classical and effective dimension theory for general covering classes Φ, extends point-to-set principles to this setting, and supplies an alternative proof of an existing theorem on Cantor covering faithfulness. The new sequence-construction technique for Kolmogorov complexity conditions and the explicit corollaries for classical dimensions are explicit strengths. The equivalence result clarifies when restricted coverings preserve dimension in both settings.

minor comments (3)

[§2] §2 (definitions): the notation for the covering class Φ is introduced without an immediate side-by-side comparison to the classical Hausdorff measure definition, which would aid readers unfamiliar with the Φ-generalization.
[§4.2] §4.2 (Kolmogorov characterization): the statement of the complexity bound uses an implicit constant that is not explicitly tracked in the subsequent application to the point-to-set principle; adding a parenthetical remark on the constant would improve traceability.
[Figure 1] Figure 1 (Cantor covering example): the caption does not indicate the specific sequence of partial quotients used to generate the depicted covering, making it harder to connect the figure to the faithfulness conditions derived later.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading, positive summary of the contributions, and recommendation of minor revision. No specific major comments appear in the report, so we have no individual points requiring response or revision at this time.

Circularity Check

0 steps flagged

No circularity; new definitions and proofs are self-contained

full rationale

The paper introduces constructive Φ-dimension as a new effective notion, then proves a point-to-set principle for Φ-dimension and supplies a Kolmogorov complexity characterization of the new notion. These results are applied, together with an explicit new technique for constructing sequences satisfying a Kolmogorov complexity condition, to derive equivalence of faithfulness notions for Cantor coverings and the necessary/sufficient conditions. No quoted step reduces a claimed prediction or theorem to a fitted parameter, a self-definition, or a load-bearing self-citation; the information-theoretic reproof of Albeverio et al. is presented as a derived consequence rather than an input. The derivation chain is therefore independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

Based on abstract only; the work rests on standard background from Hausdorff dimension theory and Kolmogorov complexity, plus the new definitions introduced here.

axioms (2)

standard math Standard axioms and properties of Hausdorff dimension and Φ-coverings
Invoked throughout the development of constructive Φ-dimension and point-to-set principles.
domain assumption Kolmogorov complexity provides valid characterizations for constructive dimensions
Used to characterize constructive Φ-dimension and support the faithfulness results.

invented entities (1)

constructive Φ-dimension no independent evidence
purpose: Effective version of Hausdorff Φ-dimension incorporating computability
New definition introduced to enable point-to-set principle and faithfulness analysis.

pith-pipeline@v0.9.0 · 5790 in / 1352 out tokens · 45260 ms · 2026-05-24T02:59:51.581702+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

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[1] [1]

Albeverio, Ganna Ivanenko, Mykola Lebid, and Grygori y Torbin

S. Albeverio, Ganna Ivanenko, Mykola Lebid, and Grygori y Torbin. On the Hausdorﬀ di- mension faithfulness and the Cantor series expansion. Methods of Functional Analysis and Topology, 26(4):298–310, 2020

work page 2020

[2] [2]

Fractal properti es of singular probability distributions with independent Q∗-digits

Sergio Albeverio and Grygoriy Torbin. Fractal properti es of singular probability distributions with independent Q∗-digits. Bull. Sci. Math. , 129(4):356–367, 2005. doi:10.1016/j.bulsci.2004.12.001

work page doi:10.1016/j.bulsci.2004.12.001 2005

[3] [3]

A. S. Besicovitch. On existence of subsets of ﬁnite measu re of sets of inﬁnite measure. Indag. Math., 14:339–344, 1952. Nederl. Akad. Wetensch. Proc. Ser. A 55

work page 1952

[4] [4]

Uber die einfachen zahlensysteme, zeit

George Cantor. Uber die einfachen zahlensysteme, zeit. f¨ ur math. 14 (1869), 121-128. Coll. Papers, Berlin , pages 35–53, 1932

work page 1932

[5] [5]

Downey and Denis R

Rodney G. Downey and Denis R. Hirschfeldt. Algorithmic randomness and com- plexity. Theory and Applications of Computability. Springer, New Y ork, 2010. doi:10.1007/978-0-387-68441-3 . 20

work page doi:10.1007/978-0-387-68441-3 2010

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Downey and Denis R

Rodney G. Downey and Denis R. Hirschfeldt. Algorithmic randomness and complexity . Springer Science & Business Media, 2010

work page 2010

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Fractal geometry

Kenneth Falconer. Fractal geometry. John Wiley & Sons, Inc., Hoboken, NJ, second edition,

work page

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doi:10.1002/0470013850

Mathematical foundations and applications. doi:10.1002/0470013850

work page doi:10.1002/0470013850

[9] [9]

Hausdorﬀ

F. Hausdorﬀ. Dimension und ¨ ausseres Mass. Mathematische Annalen , 79:157–179, 1919

work page 1919

[10] [10]

Hitchcock and Elvira Mayordomo

John M. Hitchcock and Elvira Mayordomo. Base invariance of feasible dimension. Inform. Process. Lett., 113(14-16):546–551, 2013. doi:10.1016/j.ipl.2013.04.004

work page doi:10.1016/j.ipl.2013.04.004 2013

[11] [11]

M. Kh. ¯Ibrag ¯ ım and G. M. Torb ¯ ın. On a probabilistic approach to th e DP-transformations and faithfulness of covering systems for calculating the Ha usdorﬀ-Besicovitch dimension. Teor. ˘ Imov ¯ ır. Mat. Stat., (92):28–40, 2015. doi:10.1090/tpms/980

work page doi:10.1090/tpms/980 2015

[12] [12]

An introduction to Kolmogorov complexity and its ap- plications

Ming Li and Paul Vit´ anyi. An introduction to Kolmogorov complexity and its ap- plications. Texts in Computer Science. Springer, New York, third editi on, 2008. doi:10.1007/978-0-387-49820-1

work page doi:10.1007/978-0-387-49820-1 2008

[13] [13]

J. H. Lutz. Dimensions of individual strings and sequen ces. Information and Computation , 187(1):49–79, 2003

work page 2003

[14] [14]

Jack H. Lutz. Dimension in complexity classes. SIAM J. Comput. , 32(5):1236–1259, 2003. doi:10.1137/S0097539701417723

work page doi:10.1137/s0097539701417723 2003

[15] [15]

Lutz and Neil Lutz

Jack H. Lutz and Neil Lutz. Algorithmic information, pl ane kakeya sets, and conditional dimension. ACM Trans. Comput. Theory , 10(2):7:1–7:22, 2018. doi:10.1145/3201783

work page doi:10.1145/3201783 2018

[16] [16]

Lutz and Neil Lutz

Jack H. Lutz and Neil Lutz. Who asked us? how the theory of computing answers questions about analysis. In Ding-Zhu Du and Jie Wang, editors, Complexity and Approximation - In Memory of Ker-I Ko , volume 12000 of Lecture Notes in Computer Science , pages 48–56. Springer, 2020. doi:10.1007/978-3-030-41672-0\_4

work page doi:10.1007/978-3-030-41672-0 2020

[17] [17]

Lutz, Neil Lutz, and Elvira Mayordomo

Jack H. Lutz, Neil Lutz, and Elvira Mayordomo. Extendin g the reach of the point-to-set prin- ciple. In Petra Berenbrink and Benjamin Monmege, editors, 39th International Symposium on Theoretical Aspects of Computer Science, STACS 2022, March 15- 18, 2022, Marseille, France (Virtual Conference) , volume 219 of LIPIcs, pages 48:1–48:14. Schloss Dagstuhl - ...

work page doi:10.4230/lipics.stacs.2022.48 2022

[18] [18]

Lutz and Elvira Mayordomo

Jack H. Lutz and Elvira Mayordomo. Dimensions of points in self-similar fractals. SIAM Journal on Computing , 38:1080–1112, 2008

work page 2008

[19] [19]

Lutz and Klaus Weihrauch

Jack H. Lutz and Klaus Weihrauch. Connectivity propert ies of dimension level sets. MLQ Math. Log. Q. , 54(5):483–491, 2008. doi:10.1002/malq.200710060

work page doi:10.1002/malq.200710060 2008

[20] [20]

Fractal intersections and products via algo rithmic dimension

Neil Lutz. Fractal intersections and products via algo rithmic dimension. ACM Trans. Comput. Theory, 13(3):Art. 14, 15, 2021. doi:10.1145/3460948

work page doi:10.1145/3460948 2021

[21] [21]

Neil Lutz and D. M. Stull. Bounding the dimension of poin ts on a line. Inform. and Comput. , 275:104601, 15, 2020. doi:10.1016/j.ic.2020.104601. 21

work page doi:10.1016/j.ic.2020.104601 2020

[22] [22]

Neil Lutz and Donald M. Stull. Projection theorems usin g eﬀective dimension. In 43rd Interna- tional Symposium on Mathematical Foundations of Computer Sc ience, volume 117 of LIPIcs. Leibniz Int. Proc. Inform. , pages Art. No. 71, 15. Schloss Dagstuhl. Leibniz-Zent. Inf orm., Wadern, 2018

work page 2018

[23] [23]

A Kolmogorov complexity characteri zation of constructive Hausdorﬀ di- mension

Elvira Mayordomo. A Kolmogorov complexity characteri zation of constructive Hausdorﬀ di- mension. Inform. Process. Lett. , 84(1):1–3, 2002. doi:10.1016/S0020-0190(02)00343-5

work page doi:10.1016/s0020-0190(02)00343-5 2002

[24] [24]

Eﬀective hausdorﬀ dimension in gener al metric spaces

Elvira Mayordomo. Eﬀective hausdorﬀ dimension in gener al metric spaces. Theory Comput. Syst., 62(7):1620–1636, 2018. doi:10.1007/s00224-018-9848-3

work page doi:10.1007/s00224-018-9848-3 2018

[25] [25]

An eﬀective ergodic theorem and so me applications

Satyadev Nandakumar. An eﬀective ergodic theorem and so me applications. In STOC’08, pages 39–44. ACM, New York, 2008. doi:10.1145/1374376.1374383

work page doi:10.1145/1374376.1374383 2008

[26] [26]

Eﬀec tive continued fraction dimen- sion versus eﬀective Hausdorﬀ dimension of reals

Satyadev Nandakumar, Akhil S, and Prateek Vishnoi. Eﬀec tive continued fraction dimen- sion versus eﬀective Hausdorﬀ dimension of reals. In 48th International Symposium on Mathematical Foundations of Computer Science , volume 272 of LIPIcs. Leibniz Int. Proc. Inform., pages Paper No. 70, 15. Schloss Dagstuhl. Leibniz-Zent. In form., Wadern, 2023. doi:10.4...

work page doi:10.4230/lipics.mfcs.2023.70 2023

[27] [27]

Randomness a nd eﬀective dimension of contin- ued fractions

Satyadev Nandakumar and Prateek Vishnoi. Randomness a nd eﬀective dimension of contin- ued fractions. In 45th International Symposium on Mathematical Foundations o f Computer Science, volume 170 of LIPIcs. Leibniz Int. Proc. Inform. , pages Art. No. 73, 13. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2020

work page 2020

[28] [28]

On continued fraction random- ness and normality

Satyadev Nandakumar and Prateek Vishnoi. On continued fraction random- ness and normality. Information and Computation , 285(part B):104876, 2022. URL: https://www.sciencedirect.com/science/article/pii/S0890540122000189, doi:https://doi.org/10.1016/j.ic.2022.104876

work page doi:10.1016/j.ic.2022.104876 2022

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