Geodesic
Approximation by $\Omega$-continued fractions
Čebyševskij sbornik, Tome 14 (2013) no. 4, pp. 95-100

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Let $x\in (0,1)$ be a real number, $x=[0;\varepsilon_1/b_1,\ldots,\varepsilon_1/b_n,\ldots]$ be its expansion in $\Omega$-continued fraction. Let $A_n/B_n$ be its nth convergent and $\Upsilon_n=\Upsilon_n(x)=B^2_n|x -A_n/B_n|$. In this note we prove the analog of the classical theorems by Borel and Hurwitz on the quality of the approximations for $\Omega$-continued fractions: $\min(\Upsilon_{n-1}, \Upsilon_{n},\Upsilon_{n+1})\le 1/\sqrt{5}$. The result is best possible.
Keywords: continued fractions, semi-regular continued fractions, approximation coefficients, Vahlen's theorem, $\Omega$-continued fraction expansion, analogue of Borel's theorem. 
@article{CHEB_2013_14_4_a5,
     author = {O. A. Gorkusha},
     title = {Approximation by $\Omega$-continued fractions},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {95--100},
     publisher = {mathdoc},
     volume = {14},
     number = {4},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2013_14_4_a5/}
}
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O. A. Gorkusha. Approximation by $\Omega$-continued fractions. Čebyševskij sbornik, Tome 14 (2013) no. 4, pp. 95-100. http://geodesic.mathdoc.fr/item/CHEB_2013_14_4_a5/