Diophantine inequalities and irrationality measures for certain transcendental numbers
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 \(\alpha\).
scientific article; zbMATH DE number 1785888