γ and e γ to 20700D and Their Regular Continued Fractions to 20000 Partial Quotients.

AbstractWe establish measures of non-quadraticity and transcendence measures for real numbers whose sequence of partial quotients has sublinear block complexity. The main new ingredient is an improvement of Liouville’s inequality giving a lower bound for the distance between two distinct quadratic real numbers. Furthermore, we discuss the gap between Mahler’s exponent w2 and Koksma’s exponent w*2.

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Renewal-type limit theorem for continued fractions with even partial quotients

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385708000825 ◽

2009 ◽

Vol 29 (5) ◽

pp. 1451-1478 ◽

Cited By ~ 3

Author(s):

FRANCESCO CELLAROSI

Keyword(s):

Limit Theorem ◽

Continued Fractions ◽

Continued Fraction ◽

Type Theorem ◽

Natural Extension ◽

Gauss Map ◽

Continued Fraction Expansion ◽

Special Flow ◽

Partial Quotients ◽

Continued Fraction Expansions

AbstractWe prove the existence of the limiting distribution for the sequence of denominators generated by continued fraction expansions with even partial quotients, which were introduced by Schweiger [Continued fractions with odd and even partial quotients. Arbeitsberichte Math. Institut Universtät Salzburg4 (1982), 59–70; On the approximation by continues fractions with odd and even partial quotients. Arbeitsberichte Math. Institut Universtät Salzburg1–2 (1984), 105–114] and studied also by Kraaikamp and Lopes [The theta group and the continued fraction expansion with even partial quotients. Geom. Dedicata59(3) (1996), 293–333]. Our main result is proven following the strategy used by Sinai and Ulcigrai [Renewal-type limit theorem for the Gauss map and continued fractions. Ergod. Th. & Dynam. Sys.28 (2008), 643–655] in their proof of a similar renewal-type theorem for Euclidean continued fraction expansions and the Gauss map. The main steps in our proof are the construction of a natural extension of a Gauss-like map and the proof of mixing of a related special flow.

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CONTINUED FRACTIONS WITH BOUNDED PARTIAL QUOTIENTS

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s001309150000119x ◽

2002 ◽

Vol 45 (3) ◽

pp. 653-671 ◽

Cited By ~ 6

Author(s):

J. L. Davison

Keyword(s):

Rate Of Convergence ◽

Continued Fractions ◽

Continued Fraction ◽

Irrational Number ◽

Subject Classification ◽

Specific Sequence ◽

Secondary 11B37 ◽

Quadratic Irrational ◽

Infinite Word ◽

Partial Quotients

AbstractPrecise bounds are given for the quantity$$ L(\alpha)=\frac{\limsup_{m\rightarrow\infty}(1/m)\ln q_m}{\liminf_{m\rightarrow\infty}(1/m)\ln q_m}, $$where $(q_m)$ is the classical sequence of denominators of convergents to the continued fraction $\alpha=[0,u_1,u_2,\dots]$ and $(u_m)$ is assumed bounded, with a distribution.If the infinite word $\bm{u}=u_1u_2\dots$ has arbitrarily large instances of segment repetition at or near the beginning of the word, then we quantify this property by means of a number $\gamma$, called the segment-repetition factor.If $\alpha$ is not a quadratic irrational, then we produce a specific sequence of quadratic irrational approximations to $\alpha$, the rate of convergence given in terms of $L$ and $\gamma$. As an application, we demonstrate the transcendence of some continued fractions, a typical one being of the form $[0,u_1,u_2,\dots]$ with $u_m=1+\lfloor m\theta\rfloor\Mod n$, $n\geq2$, and $\theta$ an irrational number which satisfies any of a given set of conditions.AMS 2000 Mathematics subject classification: Primary 11A55. Secondary 11B37

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