Diophantine inequalities and irrationality measures for certain transcendental numbers
type: Item
instance of: scholarly article
title: Diophantine inequalities and irrationality measures for certain transcendental numbers
zbMATH Open document ID: 1012.11057
author: Edward B. Burger
published in: Indian Journal of Pure \& Applied Mathematics
publication date: 2003-06-16T00:00:00Z
review text: Let \(\alpha\) be complex number. Let \(\beta\) be an algebraic number of degree \(N\), whose height does not exceed \(H\). Let \(\omega(N,H)\) be defined by \(|\alpha-\beta|=H^{-N\omega(N,H)}\). If there exists an \(N\) such that \(\omega(N,H)\) is unbounded, then \(\alpha\) is called a \(U\)-number. If \(N\) is the smallest such integer for which \(\omega(N,H)\) is unbounded, then \(\alpha\) is called a \(U_N\)-number. Let \(\mathcal S\) be any nonempty set of nonnegative integers. The author and \textit{T. Struppeck} proved that for any \(\mathcal S\) there exist uncountably many \(U_2\)-numbers \(\alpha\) such that \((\alpha+k)^2\) is a Liouville number for all \(k\in\mathcal S\) in [Mathematika 40, 215-225 (1993; Zbl 0796.11026)]. This paper studies the behaviour of \((\alpha+k)^2\) for \(k\in\mathcal S\) and \((\alpha+m)^2\) for natural numbers \(m\not\in\mathcal S\), describing Diophantine inequalities and irrationality measures for \(U_2\)-numbers \(\alpha\).
Mathematics Subject Classification ID: 11J25
Mathematics Subject Classification ID: 11J82
zbMATH DE Number: 1785888
zbMATH Keywords: Diophantine inequalities
zbMATH Keywords: irrationality measures
zbMATH Keywords: transcendental numbers
zbMATH Keywords: Liouville number
zbMATH Keywords: \(U\)-number
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1785888