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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2012, Volume 276, Pages 198–212 (Mi tm3366)  

Diophantine approximation generalized

Ladislav Mišíka, Oto Strauchb

a Department of Mathematics, University of Ostrava, Ostrava, Czech Republic
b Mathematical Institute, Slovak Academy of Sciences, Bratislava, Slovakia
References:
Abstract: In this paper we study the set of $x\in[0,1]$ for which the inequality $|x-x_n|<z_n$ holds for infinitely many $n=1,2,\dots$. Here $x_n\in[0,1)$ and $z_n>0$, $z_n\to0$, are sequences. In the first part of the paper we summarize known results. In the second part, using the theory of distribution functions of sequences, we find the asymptotic density of $n$ for which $|x-x_n|<z_n$, where $x$ is a discontinuity point of some distribution function of $x_n$. Generally, we also prove, for an arbitrary sequence $x_n$, that there exists $z_n$ such that the density of $n=1,2,\dots$, $x_n\to x$, is the same as the density of $n=1,2,\dots$, $|x-x_n|<z_n$, for $x\in[0,1]$. Finally we prove, using the longest gap $d_n$ in the finite sequence $x_1,x_2,\dots,x_n$, that if $d_n\le z_n$ for all $n$, $z_n\to0$, and $z_n$ is non-increasing, then $|x-x_n|<z_n$ holds for infinitely many $n$ and for almost all $x\in[0,1]$.
Received in August 2011
English version:
Proceedings of the Steklov Institute of Mathematics, 2012, Volume 276, Pages 193–207
DOI: https://doi.org/10.1134/S0081543812010166
Bibliographic databases:
Document Type: Article
UDC: 511.36+511.21
Language: English
Citation: Ladislav Mišík, Oto Strauch, “Diophantine approximation generalized”, Number theory, algebra, and analysis, Collected papers. Dedicated to Professor Anatolii Alekseevich Karatsuba on the occasion of his 75th birthday, Trudy Mat. Inst. Steklova, 276, MAIK Nauka/Interperiodica, Moscow, 2012, 198–212; Proc. Steklov Inst. Math., 276 (2012), 193–207
Citation in format AMSBIB
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\by Ladislav~Mi{\v s}{\'\i}k, Oto~Strauch
\paper Diophantine approximation generalized
\inbook Number theory, algebra, and analysis
\bookinfo Collected papers. Dedicated to Professor Anatolii Alekseevich Karatsuba on the occasion of his 75th birthday
\serial Trudy Mat. Inst. Steklova
\yr 2012
\vol 276
\pages 198--212
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
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    Труды Математического института имени В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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