Location of approximations of a Markoff theorem
type: Item
instance of: scholarly article
title: Location of approximations of a Markoff theorem
zbMATH Open document ID: 0714.11038
DOI: 10.1155/S0161171290000308
author: D. Kharzeev
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/46537
review text: For an irrational number x with simple continued fraction expansion \([0;a_ 1,a_ 2,...]\) let \(M_ n=a_{n+1}+[0;a_ n,...,a_ 1]+[0;a_{n+2},a_{n+3},...].\) A classical result by Hurwitz and Borel asserts \[ \max \quad \{M_{n+j} | \quad j=0,1,2,\}>\sqrt{5}\quad (n\in {\mathbb{N}}). \] The first two authors [Proc. Am. Math. Soc. 97, 19-20 (1986; Zbl 0593.10030), see also \textit{E. M. Wright}, Math. Gaz. 48, 288- 289 (1964; Zbl 0163.295)] have shown that \(\max \quad \{M_{n+j} | \quad j=0,1,2\}>\sqrt{8},\) provided \(a_{n+2}=2\). As to the third minimum in the Markov chain, it is now proved that \(\max \quad \{M_{n+j} | \quad j=0,1,4\}>\sqrt{221}/5,\) if \(a_{n+2}=2,a_{n+3}=1\).
Mathematics Subject Classification ID: 11J06
Mathematics Subject Classification ID: 11J70
zbMATH DE Number: 4175039
zbMATH Keywords: Hurwitz type theorems
zbMATH Keywords: continued fraction
zbMATH Keywords: Markov chain
reviewed by: Gerhard Ramharter
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4175039