Fractional dimension of some exceptional sets in continued fractions

Mumtaz Hussain; Rebecca Smith; Zhenliang Zhang

doi:10.5802/crmath.699

Article de recherche - Théorie des nombres

Fractional dimension of some exceptional sets in continued fractions
[Dimension fractionnaire de certains ensembles exceptionnels dans les fractions continues]

Mumtaz Hussain ¹ ; Rebecca Smith ² ; Zhenliang Zhang ³

¹ Department of Mathematical and Physical Sciences, La Trobe University, Bendigo 3552, Australia
² The University of Newcastle, Callaghan 2308, NSW, Australia
³ School of Mathematical Sciences, Chongqing Normal University, Chongqing, 401331, P. R. China

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 57-68.

Résumé
Abstract

Dans cet article, nous calculons la dimension de Hausdorff de certains ensembles exceptionnels qui émergent de contraintes spécifiques imposées aux quotients partiels des fractions continues. En particulier, nous calculons la dimension de Hausdorff des ensembles

Λ_{1} = \{x \in (0, 1) : a_{n + 1} (x) \geq \sum_{i = 1}^{n} a_{i} (x), for all n \in ℕ\},

et

Λ_{2} = \{x \in (0, 1) : a_{n + 1} (x) \geq \sum_{i = 1}^{n} a_{i} (x), for infinitely many n \in ℕ\} .

Nous prouvons que les dimensions de Hausdorff de $Λ_{1}$ et $Λ_{2}$ sont respectivement $1 / 2$ et $1$ . La dimension de Hausdorff de certains autres ensembles apparentés, obtenus en considérant différents taux de croissance plus rapides tels que le remplacement du taux de croissance des sommes de quotients partiels par le produit des quotients partiels dans les ensembles ci-dessus, est également calculée avec les bornes de dimension $1 / 3$ et au moins $2 / 3$ .

In this paper, we calculate the Hausdorff dimension of some exceptional sets that emerge from specific constraints imposed on the partial quotients of continued fractions. In particular, we calculate the Hausdorff dimension of the sets

Λ_{1} = \{x \in (0, 1) : a_{n + 1} (x) \geq \sum_{i = 1}^{n} a_{i} (x), for all n \in ℕ\},

and

Λ_{2} = \{x \in (0, 1) : a_{n + 1} (x) \geq \sum_{i = 1}^{n} a_{i} (x), for infinitely many n \in ℕ\} .

We prove that the Hausdorff dimensions of $Λ_{1}$ and $Λ_{2}$ are $1 / 2$ and $1$ respectively. The Hausdorff dimension of some other related sets, obtained by considering different faster growth rates such as replacing the growth rate of sums of partial quotients with the product of partial quotients in the above sets, is also calculated with the dimension bounds $1 / 3$ and at least $2 / 3$ .

Reçu le : 2024-04-12
Révisé le : 2024-06-24
Accepté le : 2024-11-04
Publié le : 2025-03-06

DOI : 10.5802/crmath.699

Classification : 11K55, 28A80
Keywords: Continued fractions, growth rate, Hausdorff dimension
Mots-clés : Fractions continues, ensembles exceptionnels, dimension de Hausdorff

Affiliations des auteurs :

Mumtaz Hussain ¹ ; Rebecca Smith ² ; Zhenliang Zhang ³

¹ Department of Mathematical and Physical Sciences, La Trobe University, Bendigo 3552, Australia
² The University of Newcastle, Callaghan 2308, NSW, Australia
³ School of Mathematical Sciences, Chongqing Normal University, Chongqing, 401331, P. R. China

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2025__363_G1_57_0,
     author = {Mumtaz Hussain and Rebecca Smith and Zhenliang Zhang},
     title = {Fractional dimension of some exceptional sets in continued fractions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {57--68},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {363},
     year = {2025},
     doi = {10.5802/crmath.699},
     language = {en},
}

TY  - JOUR
AU  - Mumtaz Hussain
AU  - Rebecca Smith
AU  - Zhenliang Zhang
TI  - Fractional dimension of some exceptional sets in continued fractions
JO  - Comptes Rendus. Mathématique
PY  - 2025
SP  - 57
EP  - 68
VL  - 363
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.699
LA  - en
ID  - CRMATH_2025__363_G1_57_0
ER  -

%0 Journal Article
%A Mumtaz Hussain
%A Rebecca Smith
%A Zhenliang Zhang
%T Fractional dimension of some exceptional sets in continued fractions
%J Comptes Rendus. Mathématique
%D 2025
%P 57-68
%V 363
%I Académie des sciences, Paris
%R 10.5802/crmath.699
%G en
%F CRMATH_2025__363_G1_57_0

Mumtaz Hussain; Rebecca Smith; Zhenliang Zhang. Fractional dimension of some exceptional sets in continued fractions. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 57-68. doi : 10.5802/crmath.699. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.699/

Bibliographie
Cité par

[1] Philip Bos; Mumtaz Hussain; David Simmons The generalised Hausdorff measure of sets of Dirichlet non-improvable numbers, Proc. Am. Math. Soc., Volume 151 (2023) no. 5, pp. 1823-1838 | DOI | MR | Zbl

[2] Kenneth Falconer Fractal geometry. Mathematical foundations and applications, John Wiley & Sons, 2014, xxx+368 pages | MR

[3] Ai-Hua Fan; Ling-Min Liao; Bao-Wei Wang; Jun Wu On Khintchine exponents and Lyapunov exponents of continued fractions, Ergodic Theory Dyn. Syst., Volume 29 (2009) no. 1, pp. 73-109 | DOI | MR | Zbl

[4] De Jun Feng; Jun Wu; Jyh-Ching Liang; Shiojenn Tseng Appendix to the paper by T. Łuczak—a simple proof of the lower bound: “On the fractional dimension of sets of continued fractions”, Mathematika, Volume 44 (1997) no. 1, pp. 54-55 | DOI | MR

[5] Dmitry Gayfulin; Nikita Shulga Diophantine properties of fixed points of Minkowski question mark function, Acta Arith., Volume 195 (2020) no. 4, pp. 367-382 | DOI | MR | Zbl

[6] Irving John Good The fractional dimensional theory of continued fractions, Proc. Camb. Philos. Soc., Volume 37 (1941), pp. 199-228 | DOI | MR | Zbl

[7] Lingling Huang; Jun Wu; Jian Xu Metric properties of the product of consecutive partial quotients in continued fractions, Isr. J. Math., Volume 238 (2020) no. 2, pp. 901-943 | DOI | MR | Zbl

[8] Mumtaz Hussain; Dmitry Kleinbock; Nick Wadleigh; Bao-Wei Wang Hausdorff measure of sets of Dirichlet non-improvable numbers, Mathematika, Volume 64 (2018) no. 2, pp. 502-518 | DOI | MR | Zbl

[9] Mumtaz Hussain; Bixuan Li; Nikita Shulga Hausdorff dimension analysis of sets with the product of consecutive vs single partial quotients in continued fractions, Discrete Contin. Dyn. Syst., Volume 44 (2024) no. 1, pp. 154-181 | DOI | MR | Zbl

[10] Mumtaz Hussain; Nikita Shulga Metrical properties of exponentially growing partial quotients (To appear in Forum Math.) | DOI

[11] Marius Iosifescu; Cor Kraaikamp Metrical theory of continued fractions, Mathematics and its Applications, 547, Kluwer Academic Publishers, 2002, xx+383 pages | DOI | MR

[12] Vojtěch Jarník A contribution to the metric theory of Diophantine approximations, Prace Mat.-Fiz., Volume 36 (1929), pp. 91-106

[13] Aleksandr Yakovlevich Khinchin Continued fractions, University of Chicago Press, 1964, xi+95 pages | MR

[14] Dmitry Kleinbock; Nick Wadleigh A zero-one law for improvements to Dirichlet’s Theorem, Proc. Am. Math. Soc., Volume 146 (2018) no. 5, pp. 1833-1844 | DOI | MR | Zbl

[15] Bing Li; Bao-Wei Wang; Jun Wu; Jian Xu The shrinking target problem in the dynamical system of continued fractions, Proc. Lond. Math. Soc., Volume 108 (2014) no. 1, pp. 159-186 | DOI | MR | Zbl

[16] Tomasz Łuczak On the fractional dimension of sets of continued fractions, Mathematika, Volume 44 (1997) no. 1, pp. 50-53 | DOI | MR | Zbl

[17] Bo Tan; Qinglong Zhou The relative growth rate for partial quotients in continued fractions, J. Math. Anal. Appl., Volume 478 (2019) no. 1, pp. 229-235 | DOI | MR | Zbl

[18] Bao-Wei Wang; Jun Wu Hausdorff dimension of certain sets arising in continued fraction expansions, Adv. Math., Volume 218 (2008) no. 5, pp. 1319-1339 | DOI | MR | Zbl

[19] Bao-Wei Wang; Jun Wu; Jian Xu A generalization of the Jarník-Besicovitch theorem by continued fractions, Ergodic Theory Dyn. Syst., Volume 36 (2016) no. 4, pp. 1278-1306 | DOI | MR | Zbl

[20] Jun Wu A remark on the growth of the denominators of convergents, Monatsh. Math., Volume 147 (2006) no. 3, pp. 259-264 | DOI | MR | Zbl

Cité par Sources :

Commentaires - Politique