On Diophantine approximations of dependent quantities in the \(p\)-adic case
type: Item
instance of: scholarly article
title: On Diophantine approximations of dependent quantities in the \(p\)-adic case
zbMATH Open document ID: 1037.11047
DOI: 10.1023/A:1022165815830
author: È. I. Kovalevskaya
author: Victor Beresnevich
published in: Mathematical Notes
publication date: 2003-06-15T00:00:00Z
review text: Let \(f:\mathbb Z_p\rightarrow\mathbb Z_p\) be a normal function in the sense of Mahler, and let \(\mu\) be the Haar measure on \(\mathbb Q_p\) normalized so that \(\mu(\mathbb Z_p)=p\). Suppose that \(f''(x)\neq0\) almost everywhere, and let \(\psi:\mathbb N\rightarrow\mathbb R^+\) be monotonically decreasing. Let \(\mathcal L_f(\psi)\) be the set of \(x\in\mathbb Z_p\) for which \[ | a_2f(x)+a_1x+a_0|_p<\psi(\max| a_i|) \] has infinitely many solutions \((a_0,a_1,a_2)\in\mathbb Z^3\). It is proved that \(\mu(\mathcal L_f(\psi))= 0\) or \(p\) according as \(\sum_{h=1}^\infty h^2\psi(h)\) converges or diverges. The convergence case has been proved earlier (assuming that \(h^2\psi(h)\) is decreasing) by the second author [Dokl. Nats. Akad. Nauk Belarusi 44, No. 2, 28--30 (2000; Zbl 1177.11067)].
Mathematics Subject Classification ID: 11J61
Mathematics Subject Classification ID: 11J83
Mathematics Subject Classification ID: 11K60
Mathematics Subject Classification ID: 11K41
zbMATH DE Number: 1928358
zbMATH Keywords: Diophantine approximation
zbMATH Keywords: plane curve
zbMATH Keywords: \(p\)-adic integers
zbMATH Keywords: Mahler normal function
zbMATH Keywords: Haar measure
reviewed by: Veikko Ennola
MaRDI profile type: Publication
full work available at URL: a:1022165815830
OpenAlex ID: W33304581
description: scientific article; zbMATH DE number 1928358