On Mahler's \(p\)-adic \(U_m\)-numbers
type: Item
instance of: scholarly article
zbMATH Open document ID: 1282.11077
DOI: 10.1017/S0004972712000664
author: Gülcan Kekeç
published in: Bulletin of the Australian Mathematical Society
publication date: 2013-09-11T00:00:00Z
Mathematics Subject Classification ID: 11J61
Mathematics Subject Classification ID: 11J82
zbMATH DE Number: 6207401
zbMATH Keywords: Mahler's classification of \(p\)-adic numbers
zbMATH Keywords: \(p\)-adic \(U\)-number
zbMATH Keywords: transcendence measure
MaRDI profile type: Publication
cites work: On \(U_m\)-numbers with small transcendence measure
title: On Mahler's \(p\)-adic \(U_m\)-numbers
review text: 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.
reviewed by: Peter Bundschuh
description: scientific article; zbMATH DE number 6207401