Engel连分数展式中例外集的Hausdorff维数

吕美英, 谢婧

数学学报 ›› 2022, Vol. 65 ›› Issue (6) : 1003-1008.

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数学学报 ›› 2022, Vol. 65 ›› Issue (6) : 1003-1008. DOI: 10.12386/A20210058
论文

Engel连分数展式中例外集的Hausdorff维数

    吕美英, 谢婧
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Hausdorff Dimension of the Exceptional Set in Engel Continued Fractions

    Mei Ying LÜ, Jing XIE
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摘要

对于任意的实数x∈（0，1），都有其唯一的Engel连分数展式.本文主要研究了部分商的对数构成的序列以非线性速度增长的例外集，给出了相关例外集Hausdorff维数的精确刻画.

Abstract

For any real number x∈(0,1), there exists a unique Engel continued fractions of x. In this paper, we mainly discuss the exceptional set which the logarithms of the partial quotients grow with non-linear rate. We completely characterize the Hausdorff dimension of the relevant exceptional set.

关键词

Engel连分数 / 例外集 / Hausdorff维数

Key words

Engel continued fractions / exceptional sets / Hausdorff dimensions

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吕美英, 谢婧. Engel连分数展式中例外集的Hausdorff维数. 数学学报, 2022, 65(6): 1003-1008 https://doi.org/10.12386/A20210058
Mei Ying LÜ, Jing XIE. Hausdorff Dimension of the Exceptional Set in Engel Continued Fractions. Acta Mathematica Sinica, Chinese Series, 2022, 65(6): 1003-1008 https://doi.org/10.12386/A20210058

参考文献

[1] Falconer K., Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons, Ltd., Chichester, 1990.
[2] Galambos J., Representations of Real Numbers by Infinite Series, Lecture Notes in Math., Vol. 502, Springer, 1976.
[3] Hartono Y., Kraaikamp C., Schweiger F., Algebraic and ergodic properties of a new continued fraction algorithm with non-decreasing partial quotients, J. Théor. Nombres Bordeaux, 2002, 14(2): 497–516.
[4] Kraaikamp C., Wu J., On a new continued fraction expansion with non-decreasing partial quotients, Monatsh. Math., 2004, 143(4): 285–298.
[5] Shang L., Wu M., Slow growth rate of the digits in Engel expansions, Fractals, 2020, 28(3): 2050047.

基金

重庆市科委自然科学基金（cstc2018jcyjAX0277）；重庆市教委科学技术研究项目（KJQN202000531）
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