scholarly journals Dimension Estimates for Certain Sets of Infinite Complex Continued Fractions

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
J. Neunhäuserer
Keyword(s):  
Hausdorff Dimension ◽  
Continued Fractions ◽  
Gaussian Integers ◽  
Complex Continued Fractions

We prove upper and lower estimates on the Hausdorff dimension of sets of infinite complex continued fractions with finitely many prescribed Gaussian integers. Particulary we will conclude that the dimension of theses sets is not zero or two and there are such sets with dimension greater than one and smaller than one.

scholarly journals Extreme Value Theory for Hurwitz Complex Continued Fractions

Entropy ◽  
2021 ◽  
Vol 23 (7) ◽  
pp. 840
Author(s):  
Maxim Sølund Kirsebom
Keyword(s):  
Invariant Measure ◽  
Extreme Value Theory ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Value Theory ◽  
Extreme Value ◽  
Poisson Law ◽  
Complex Continued Fractions

The Hurwitz complex continued fraction is a generalization of the nearest integer continued fraction. In this paper, we prove various results concerning extremes of the modulus of Hurwitz complex continued fraction digits. This includes a Poisson law and an extreme value law. The results are based on cusp estimates of the invariant measure about which information is still limited. In the process, we obtained several results concerning the extremes of nearest integer continued fractions as well.

scholarly journals Hausdorff dimension of an exceptional set in the theory of continued fractions

Nonlinearity ◽  
2020 ◽  
Vol 33 (6) ◽  
pp. 2615-2639 ◽  
Author(s):  
Ayreena Bakhtawar ◽  
Philip Bos ◽  
Mumtaz Hussain
Keyword(s):  
Hausdorff Dimension ◽  
Continued Fractions ◽  
Exceptional Set

Kuzmin's theorem revisited

2000 ◽  
Vol 20 (2) ◽  
pp. 557-565 ◽  
Author(s):  
F. SCHWEIGER
Keyword(s):  
Convergence Rate ◽  
Continued Fractions ◽  
Complex Continued Fractions

Iosifescu pointed out that earlier attempts to generalize Kuzmin's theorem to fibred systems have a gap. In this paper a new proof is given. The convergence rate is worse, but the result applies to multi-dimensional continued fractions and complex continued fractions.

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