Hostname: page-component-5b777bbd6c-7mr9c Total loading time: 0 Render date: 2025-06-18T23:44:57.565Z Has data issue: false hasContentIssue false
  • English
  • Français

On Dirichlet non-improvable numbers and shrinking target problems

Part of: Classical measure theory Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  11 June 2025

QIAN XIAO
QIAN XIAO*
Affiliation:
School of Mathematics and Statistics, Southwest University, Chongqing 400715, P.R. China
*
e-mail: xiaoqianmath@163.com
Get access Rights & Permissions [Opens in a new window]

Abstract

In one-dimensional Diophantine approximation, the Diophantine properties of a real number are characterized by its partial quotients, especially the growth of its large partial quotients. Notably, Kleinbock and Wadleigh [Proc. Amer. Math. Soc. 146(5) (2018), 1833–1844] made a seminal contribution by linking the improvability of Dirichlet’s theorem to the growth of the product of consecutive partial quotients. In this paper, we extend the concept of Dirichlet non-improvable sets within the framework of shrinking target problems. Specifically, consider the dynamical system $([0,1), T)$ of continued fractions. Let $\{z_n\}_{n \ge 1}$ be a sequence of real numbers in $[0,1]$ and let $B> 1$. We determine the Hausdorff dimension of the following set: $ \{x\in [0,1):|T^nx-z_n||T^{n+1}x-Tz_n|<B^{-n}\text { infinitely often}\}. $

Keywords

Dirichlet non-improvable setsshrinking target problemsHausdorff dimension

MSC classification

Primary: 11K50: Metric theory of continued fractions
Secondary: 28A80: Fractals
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , First View , pp. 1 - 29
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Allen, D. and Bárány, B.. On the Hausdorff measure of shrinking target sets on self-conformal sets. Mathematika 67(4) (2021), 807–839.10.1112/mtk.12106CrossRefGoogle Scholar
Bakhtawar, A., Bos, P. and Hussain, M.. The sets of Dirichlet non-improvable numbers versus well-approximable numbers. Ergod. Th. & Dynam. Sys. 40(12) (2020), 3217–3235.10.1017/etds.2019.41CrossRefGoogle Scholar
Bakhtawar, A., Bos, P. and Hussain, M.. Hausdorff dimension of an exceptional set in the theory of continued fractions. Nonlinearity 33(6) (2020), 2615–2639.10.1088/1361-6544/ab7726CrossRefGoogle Scholar
Bárány, B. and Rams, M.. Shrinking targets on Bedford–McMullen carpets. Proc. Lond. Math. Soc. (3) 117(5) (2018), 951–995.10.1112/plms.12151CrossRefGoogle Scholar
Bishop, C. J. and Peres, Y.. Fractals in Probability and Analysis. Cambridge University Press, Cambridge, 2017.Google Scholar
Bugeaud, Y. and Wang, B.. Distribution of full cylinders and the Diophantine properties of the orbits in $\beta$ -expansions. J. Fractal Geom. 1(2) (2014), 221–241.10.4171/jfg/6CrossRefGoogle Scholar
Falconer, K.. Sets with large intersection properties. J. Lond. Math. Soc. (2) 49(2) (1994), 267–280.10.1112/jlms/49.2.267CrossRefGoogle Scholar
Good, I. J.. The fractional dimensional theory of continued fractions. Proc. Cambridge Philos. Soc. 37 (1941), 199–228.10.1017/S030500410002171XCrossRefGoogle Scholar
Hanus, P., Mauldin, R. D. and Urbański, M.. Thermodynamic formalism and multifractal analysis of conformal infinite iterated function systems. Acta Math. Hungar. 96(1–2) (2002), 27–98.10.1023/A:1015613628175CrossRefGoogle Scholar
He, Y.. A unified approach to mass transference principle and large intersection property. Adv. Math. 471 (2025), Paper no. 110267.Google Scholar
Hill, R. and Velani, S.. Metric Diophantine approximation in Julia sets of expanding rational maps. Publ. Math. Inst. Hautes Etudes Sci. 85 (1997), 193–216.10.1007/BF02699537CrossRefGoogle Scholar
Hill, R. and Velani, S.. The shrinking target problem for matrix transformations of tori. J. Lond. Math. Soc. (2) 60(2) (1999), 381–398.10.1112/S0024610799007681CrossRefGoogle Scholar
Huang, L., Wu, J. and Xu, J.. Metric properties of the product of consecutive partial quotients in continued fractions. Israel J. Math. 238(2) (2020), 901–943.10.1007/s11856-020-2049-1CrossRefGoogle Scholar
Hussain, M., Li, B. and Shulga, N.. Hausdorff dimension analysis of sets with the product of consecutive vs single partial quotients in continued fractions. Discrete Contin. Dyn. Syst. 44(1) (2024), 154–181.10.3934/dcds.2023099CrossRefGoogle Scholar
Hussain, M. and Shulga, N.. Metrical properties of exponentially growing partial quotients. Forum Math. doi:10.1515/forum-2024-0007. Published online 30 November 2024.CrossRefGoogle Scholar
Iosifescu, M. and Kraaikamp, C.. Metrical Theory of Continued Fractions (Mathematics and Its Applications, 547). Kluwer Academic Publishers, Dordrecht, 2002.10.1007/978-94-015-9940-5CrossRefGoogle Scholar
Khintchine, A. Y.. Continued Fractions. P. Noordhoff, Groningen, The Netherlands, 1963.Google Scholar
Kleinbock, D. and Wadleigh, N.. A zero-one law for improvements to Dirichlet’s theorem. Proc. Amer. Math. Soc. 146(5) (2018), 1833–1844.10.1090/proc/13685CrossRefGoogle Scholar
Koivusalo, H., Liao, L. and Rams, M.. Path-dependent shrinking targets in generic affine iterated function systems. Preprint, 2022, arXiv:2210.05362.Google Scholar
Li, B., Liao, L., Velani, S. and Zorin, E.. The shrinking target problem for matrix transformations of tori: revisiting the standard problem. Adv. Math. 421 (2023), Paper no. 108994.10.1016/j.aim.2023.108994CrossRefGoogle Scholar
Li, B., Wang, B., Wu, J. and Xu, J.. The shrinking target problem in the dynamical system of continued fractions. Proc. Lond. Math. Soc. (3) 108(1) (2014), 159–186.10.1112/plms/pdt017CrossRefGoogle Scholar
Li, B., Wang, B. and Xu, J.. Hausdorff dimension of Dirichlet non-improvable set versus well-approximable set. Ergod. Th. & Dynam. Sys. 43(8) (2023), 2707–2731.10.1017/etds.2022.51CrossRefGoogle Scholar
Łuczak, T.. On the fractional dimension of sets of continued fractions. Mathematika 44(1) (1997), 50–53.10.1112/S0025579300011955CrossRefGoogle Scholar
Mauldin, D. and Urbański, M.. Dimensions and measures in infinite iterated function systems. Proc. Lond. Math. Soc. (3) 73(1) (1996), 105–154.10.1112/plms/s3-73.1.105CrossRefGoogle Scholar
Mauldin, D. and Urbański, M.. Conformal iterated function systems with applications to the geometry of continued fractions. Trans. Amer. Math. Soc. 351(12) (1999), 4995–5025.10.1090/S0002-9947-99-02268-0CrossRefGoogle Scholar
Wang, B. and Wu, J.. Hausdorff dimension of certain sets arising in continued fraction expansions. Adv. Math. 218(5) (2008), 1319–1339.10.1016/j.aim.2008.03.006CrossRefGoogle Scholar