Approximation by algebraic numbers (Q1587415)

Let \(m\geq 1\) and \(d\geq 2\) be integers, \(\varepsilon\), \(\lambda\) real numbers with \(0<\varepsilon<\lambda<d\) and \(\theta\) an algebraic number of degree \(>d\). There exists a real number \(H_{0}\) with the following property: For each \(H\geq H_{0}\) there is an algebraic number \(\alpha\) of degree \(d\) and usual height \(H(\alpha)\) with \[ H^{1-(\varepsilon/\lambda)}\leq H(\alpha)\leq H \quad\text{ and }\quad |\theta-\alpha|\leq H(\alpha)^{-d-1+\varepsilon}. \] This refines earlier results of \textit{E. Wirsing} [J. Reine Angew. Math. 206, 67-77 (1961; Zbl 0097.03503)] and \textit{W. M. Schmidt} [Enseign. Math., II. Sér. 17, 187--253 (1971; Zbl 0226.10033)] in two ways: the lower bound for \(H(\alpha)\) is new, as well as the information that \(\alpha \) has degree \(d\) (and not merely \(\leq d\)). A similar result is shown to be true for almost all real numbers \(\theta\) (for Lebesgue measure). Also for almost all real (resp. complex) numbers \(\theta\), the author refines the result of \textit{M.~Laurent} and \textit{D.~Roy} [Ann. Inst. Fourier 49, No. 1, 27--55 (1999; Zbl 0923.11105)] on the existence of algebraic approximation by algebraic numbers: He provides lower bounds for the degree as well as for the height of the approximants. He shows that such statements do not hold for \(U\)-numbers. Further related results are established, including a refinement of an estimate due to \textit{E.~Bombieri} and \textit{J.~Mueller} [Mich. Math. J. 33, 83--93 (1986; Zbl 0593.10031)] providing an effective improvement of Liouville's inequality.