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Mat. Zametki, 1978, Volume 23, Issue 1, Pages 3–26 (Mi mz8113)  

This article is cited in 2 scientific papers (total in 2 papers)

Rational approximations of real numbers

V. A. Ivanov

Saratov Polytechnical Institute

Abstract: For any $x\in\mathbf R$ put
$$ c(x)=\varlimsup_{t\to\infty}\min_{\substack{(p,q)\in Z\times N
q\le t}}t|qx-p|. $$
Let $[x_0;x_1,…,x_n,…]$ be an expansion of $x$ into a continued fraction and let $M=\{x\in J, \varlimsup\limits_{n\to\infty}x_n<\infty\}$. For $x\in M$ put $D(x)=c(x)/(1-c(x))$. The structure of the set $\mathfrak D=\{D(x), x\in M\}$ is studied. It is shown that
$$ \mathfrak D\cap(3+\sqrt3,(5+3\sqrt3)/2)=\{D(x^{(n,3)})\}_{n=0}^\infty\nearrow(5+3\sqrt3)/2, $$
where $x^{(n,3)}=[\overline{3;(1,2)_n,1}]$. This yields for $\mu=\infż,\mathfrak D\supset(z,+\infty)\}$ (“origin of the ray”) the following lower bound: $\mu\ge(5+3\sqrt3)/2=5,098…$. Suppose $a\in N$. Put $M(a)=\{x\in M, \varlimsup\limits_{n\to\infty}x_n=a\}$, $\mathfrak D(a)=\{D(x), x\in M(a)\}$. The smallest limit point of $\mathfrak D(a)$ $(a\ge2)$ is found. The structure of $\mathfrak D(a)$ is studied completely up to the smallest limit point and elucidated to the right of it.

Full text: PDF file (1469 kB)

English version:
Mathematical Notes, 1978, 23:1, 3–16

Bibliographic databases:

UDC: 511.7
Received: 25.03.1976

Citation: V. A. Ivanov, “Rational approximations of real numbers”, Mat. Zametki, 23:1 (1978), 3–26; Math. Notes, 23:1 (1978), 3–16

Citation in format AMSBIB
\Bibitem{Iva78}
\by V.~A.~Ivanov
\paper Rational approximations of real numbers
\jour Mat. Zametki
\yr 1978
\vol 23
\issue 1
\pages 3--26
\mathnet{http://mi.mathnet.ru/mz8113}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=476655}
\zmath{https://zbmath.org/?q=an:0406.10024|0376.10026}
\transl
\jour Math. Notes
\yr 1978
\vol 23
\issue 1
\pages 3--16
\crossref{https://doi.org/10.1007/BF01104877}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. M. V. Milentyeva, “On the dimensions of commutative subalgebras and subgroups”, J. Math. Sci., 149:2 (2008), 1135–1145  mathnet  crossref  mathscinet  zmath  elib  elib
    2. D. O. Shatskov, “On the Mean Value of the Measure of Irrationality of Real Numbers”, Math. Notes, 98:2 (2015), 301–315  mathnet  crossref  crossref  mathscinet  isi  elib
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