On Mahler's \(p\)-adic \(U_m\)-numbers (Q2847594)

Given \(\xi\in\mathbb{Q}_p\) and \(n\in\mathbb{Z}_{>0}\), let \(w_n^*(\xi)\) denote the supremum of all \(r\in\mathbb{R}\) for which there exist infinitely many \(\alpha\in\mathbb{Q}_p\), algebraic over \(\mathbb{Q}\) of degree \(\leq n\), such that \(0<|\xi-\alpha|_p<H(P_\alpha)^{-r-1}\) holds. Here \(H(P_\alpha)\) denotes the classical height of the defining minimal polynomial \(P_\alpha\) of \(\alpha\) over \(\mathbb{Z}\). With these notations, the main result of the present note reads as follows: If \(m\in\mathbb{Z}_{>1}\) and \(\varepsilon\in\mathbb{R}_+\), then there exist uncountably many \(U_m\)-numbers \(\xi\in\mathbb{Q}_p\) satisfying \(w_n^*(\xi)\leq n+m-1+ \varepsilon\) for \(n=1,\dots,m-1\).NEWLINENEWLINEThis statement can be considered as a \(p\)-adic version of the main theorem of \textit{K. Alniaçik, Y. Avci} and \textit{Y. Bugeaud} [Acta Math. Hung. 99, No. 4, 271--277 (2003; Zbl 1050.11067)] the proof of which is adapted here to the non-archimedean situation.