Extreme Value Theory for Hurwitz 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.

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Continued fractions, the Chen–Stein method and extreme value theory

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2019.64 ◽

2019 ◽

Vol 41 (2) ◽

pp. 461-470

Author(s):

ANISH GHOSH ◽

MAXIM SØLUND KIRSEBOM ◽

PARTHANIL ROY

Keyword(s):

Rate Of Convergence ◽

Extreme Value Theory ◽

Upper Bound ◽

Continued Fractions ◽

Continued Fraction ◽

Value Theory ◽

Extreme Value ◽

Real Analysis ◽

Continued Fraction Expansion ◽

Stein Method

In this work we deal with extreme value theory in the context of continued fractions using techniques from probability theory, ergodic theory and real analysis. We give an upper bound for the rate of convergence in the Doeblin–Iosifescu asymptotics for the exceedances of digits obtained from the regular continued fraction expansion of a number chosen randomly from $(0,1)$ according to the Gauss measure. As a consequence, we significantly improve the best known upper bound on the rate of convergence of the maxima in this case. We observe that the asymptotics of order statistics and the extremal point process can also be investigated using our methods.

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A Fréchet law and an Erdős–Philipp law for maximal cuspidal windings

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385712000235 ◽

2012 ◽

Vol 33 (4) ◽

pp. 1008-1028 ◽

Cited By ~ 3

Author(s):

JOHANNES JAERISCH ◽

MARC KESSEBÖHMER ◽

BERND O. STRATMANN

Keyword(s):

Extreme Value Theory ◽

Continued Fractions ◽

Geodesic Flow ◽

Fuchsian Group ◽

Value Theory ◽

Extreme Value ◽

Kleinian Groups ◽

Riemannian Surface ◽

Finitely Generated

AbstractIn this paper we establish a Fréchet law for maximal cuspidal windings of the geodesic flow on a Riemannian surface associated with an arbitrary finitely generated, essentially free Fuchsian group with parabolic elements. This result extends previous work by Galambos and Dolgopyat and is obtained by applying extreme value theory. Subsequently, we show that this law gives rise to an Erdős–Philipp law and to various generalized Khintchine-type results for maximal cuspidal windings. These results strengthen previous results by Sullivan, Stratmann and Velani for Kleinian groups, and extend earlier work by Philipp on continued fractions, which was inspired by a conjecture of Erdős.

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