Hausdorff Dimension of the Exceptional Set in Engel Continued Fractions

Mei Ying LÜ, Jing XIE

Acta Mathematica Sinica, Chinese Series ›› 2022, Vol. 65 ›› Issue (6) : 1003-1008.

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Acta Mathematica Sinica, Chinese Series ›› 2022, Vol. 65 ›› Issue (6) : 1003-1008. DOI: 10.12386/A20210058

Hausdorff Dimension of the Exceptional Set in Engel Continued Fractions

  • Mei Ying LÜ, Jing XIE
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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.

Key words

Engel continued fractions / exceptional sets / Hausdorff dimensions

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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

References

[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.
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