GROWTH RATE OF DIGITS IN A GAUSS-LIKE IFS

doi:10.1007/s10473-026-0117-2

Acta mathematica scientia,Series B ›› 2026, Vol. 46 ›› Issue (1): 293-310.doi: 10.1007/s10473-026-0117-2

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GROWTH RATE OF DIGITS IN A GAUSS-LIKE IFS

Saisai SHI¹, Bo TAN^2,*, Qinglong ZHOU³

1. Institute of Statistics and Applied Mathematics, Anhui University of Finance and Economics, Bengbu 233030, China;
2. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China;
3. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China

Received:2024-09-23 Revised:2024-12-06 Online:2026-01-25 Published:2026-05-22
Contact: * Bo TAN, E-mail: tanbo@hust.edu.cn
About author:Saisai SHI,E-mail: saisai shi@126.com;Qinglong ZHOU, E-mail: zhouql@whut.edu.cn
Supported by:
Scientific Research Project of Colleges and Universities in Anhui Province (2024AH050016). The second author was supported by the NSFC (12171172). The third author was supported by the NSFC (12201476) and the Fundamental Research Funds for the Central Universities.

Abstract

Abstract: Let $\Psi=\{\psi_{n}\}_{n\geq1}$ be an iterated function system (IFS) on [0,1] with attractor $J.$ Associated with each $x\in J,$ there is a sequence $\{\omega_{n}(x)\}_{n\geq 1}$ consisting of integers, called the digit sequence of $x,$ such that
$x=\lim_{n\rightarrow\infty}\psi_{\omega_{1}(x)}\circ\cdots\circ \psi_{\omega_{n}(x)}$(1).
We revisit the Borel-Bernstein theorem in a $d$-decaying Gauss-like IFS, and completely characterize the metrical properties of the set
$E(\Phi)=\big\{x\in J\colon \omega_{n}(x)\geq \Phi(n) \text{ for infinitely many } n\in \mathbb{N}\big\},$
where $\Phi\colon \mathbb{N}\rightarrow \mathbb{R}$ is a positive function.

Key words: Gauss-like iterated function system, Diophantine approximation, Hausdorff dimension

CLC Number:

28A80

Saisai SHI¹, Bo TAN^2,*, Qinglong ZHOU³. GROWTH RATE OF DIGITS IN A GAUSS-LIKE IFS[J].Acta mathematica scientia,Series B, 2026, 46(1): 293-310.

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References

[1] Bernstein F. Über eine Anwendung der Mengenlehre auf ein aus der Theorie der säkularen Störungen herrührendes Problem. Math Ann, 1911, ${\bf 71}$(3): 417-439
[2] Borel E. Sur un problème de probabilités relatif aux fractions continues. Math Ann, 1912, ${\bf 72}$(4): 578-584
[3] Cao C Y, Wang B W, Wu J. The growth speed of digits in infinite iterated function systems. Studia Math, 2013, ${\bf 217}$(2): 139-158
[4] Dajani K, Kraaikamp C, van der Wekken N. Ergodicity of $N$-continued fraction expansions. J Number Theory, 2013, ${\bf 133}$(9): 3183-3204
[5] Falconer K J. Fractal Geometry.Mathematical Foundations and Applications. Third edition. Chichester: John Wiley & Sons, 2014
[6] Hanus H, Mauldin R D, Urbański M. Thermodynamic formalism and multifractal analysis of conformal infinite iterated function systems. Acta Math Hungar, 2002, ${\bf 96}$(1/2): 27-98
[7] Liao L M, Rams M. Big Birkhoff sums in $d$-decaying Gauss like iterated function systems. Studia Math, 2022, ${\bf 264}$(1): 1-25
[8] Lüroth J. Ueber eine eindeutige Entwichelung von Zahlen in eine unendliche Reihe. Math Ann, 1883, ${\bf 21}$(3): 411-423
[9] Jordan T, Rams M. Increasing digit subsystems of infinite iterated function systems. Proc Amer Math Soc, 2012, ${\bf 140}$(4): 1267-1279
[10] Khintchine A Y. Einige Sätze über Kettenbrche, mit Anwendungen auf die Theorie der Diophantischen Approximationen. Math Ann, 1924, ${\bf 92}$(1/2): 115-125
[11] KhintchineA Y. {Continued Fractions}. Groningen: Dover Publications, 1963
[12] Mauldin R D, Urbański M. Dimensions and measures in infinite iterated function systems. Proc London Math Soc, 1996, ${\bf 73}$(1): 105-154
[13] Mauldin R D, Urbański M. Conformal iterated function systems with applications to the geometry of continued fractions. Trans Amer Math Soc, 1999, ${\bf 351}$(12): 4995-5025
[14] Mauldin R D, Urbański M.{Graph Directed Markov Systems. Geometry and Dynamics of Limit Sets}. Cambridge Tracts in Mathematics. 148. Cambridge: Cambridge University Press, 2003
[15] Seuret S, Wang B W. Quantitative recurrence properties in conformal iterated function systems. Adv Math, 2015, ${\bf 280}$: 472-505
[16] Shen L M. Hausdorff dimension of the set concerning with Borel-Bernstein theory in Lüroth expansions. J Korean Math Soc, 2017, ${\bf 54}$(4): 1301-1316
[17] Shi S S, Zhang Y, Liu J. Shrinking target problems in $d$-decaying Gauss-like iterated function systems. J Math Anal Appl, 2025: Article 128680
[18] Walters P.An Introduction to Ergodic Theory. Graduate Texts in Mathematics, 79. Berlin: Springer-Verlag, 1982
[19] Wang B W, Wu J. Hausdorff dimension of certain sets arising in continued fraction expansions. Adv Math, 2008, ${\bf 218}$(5): 1319-1339
[20] Zhang M J. A remark on big Birkhoff sums in $d$-decaying Gauss like iterated function systems. J Math Anal Appl, 2020, ${\bf 491}$(2): Article 124350

GROWTH RATE OF DIGITS IN A GAUSS-LIKE IFS

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