On a theorem of Wirsing in Diophantine approximation (Q2790908)

A well-known theorem of \textit{E. Wirsing} [J. Reine Angew. Math. 206, 67--77 (1961; Zbl 0097.03503)] states that for any integer \(n\geq 1\) and any real number \(\xi\) which is not algebraic of degree \(\leq n\), there exists an effectively computable constant \(c>0\), and there exist infinitely many algebraic numbers \(\alpha\) of degree \(\leq n\), satisfying NEWLINE\[NEWLINE |\xi-\alpha|\leq c H(\alpha)^{-(n+3)/2}, NEWLINE\]NEWLINE where \(H\) denotes the usual height. A famous open problem is to decide whether the exponent \(-(n+3)/2\) can be replaced by \(-n-1\). This is true at least for \(n=2\), following a result of \textit{H. Davenport} and \textit{W. M. Schmidt} [Acta Arith. 13, 169--176 (1967; Zbl 0155.09503)]. Subsequently, they refined their result [\textit{H. Davenport} and \textit{W. M. Schmidt}, Acta Arith. 14, 209--223 (1968; Zbl 0179.07303)] and proved that for \(n\geq 2\) and \(\xi\in{\mathbb{R}}\) which is not algebraic of degree \(\leq n\), there exist an effectively computable constant \(c>0\), an integer \(k\) in the range \(1\leq k\leq n-1\) and infinitely many polynomials \(P\) of degree \(n\), the roots \(\alpha_1,\dots,\alpha_n\) of which can be numbered in such a way that NEWLINE\[NEWLINE |(\xi-\alpha_1)(\xi-\alpha_2)\cdots(\xi-\alpha_k)|\leq c H(P)^{-n-1}. NEWLINE\]NEWLINE The main result of the paper under review is that for \(n\geq 2\) and \(d\) positive integers satisfying \(1\leq n\leq d-1\), for any \(\xi\in{\mathbb{R}}\) which is not algebraic of degree \(\leq n\), there exist an effectively computable constant \(c>0\), an integer \(k\) in the range \(1\leq k\leq d\) and infinitely many polynomials \(P\) of degree \(m\) with \(k\leq m\leq n\), the roots \(\alpha_1,\dots,\alpha_m\) of which can be numbered in such a way that NEWLINE\[NEWLINE |(\xi-\alpha_1)(\xi-\alpha_2)\cdots(\xi-\alpha_k)|\leq c H(P)^{-\kappa} \quad \text{with}\quad \kappa=\frac{d}{d+1} n+\frac{1}{d+1} +1. NEWLINE\]NEWLINE The proof is an extension of Wirsing's arguments.