Matematicheskie Zametki General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Mat. Zametki: Year: Volume: Issue: Page: Find

 Personal entry: Login: Password: Save password Enter Forgotten password? Register

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

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} 

• http://mi.mathnet.ru/eng/mz8113
• http://mi.mathnet.ru/eng/mz/v23/i1/p3

 SHARE:      Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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      2. D. O. Shatskov, “On the Mean Value of the Measure of Irrationality of Real Numbers”, Math. Notes, 98:2 (2015), 301–315      •  Number of views: This page: 173 Full text: 84 First page: 1 Contact us: math-net2021_10 [at] mi-ras ru Terms of Use Registration to the website Logotypes © Steklov Mathematical Institute RAS, 2021