Multiplicative \(p\)-adic approximation
type: Item
instance of: scholarly article
title: Multiplicative \(p\)-adic approximation
zbMATH Open document ID: 1496.11094
author: Dzmitry Badziahin
author: Yann Bugeaud
published in: Michigan Mathematical Journal
publication date: 2022-03-14T00:00:00Z
review text: The \(p\)-adic Littlewood conjecture of \textit{B. de Mathan} and \textit{O. Teulié} [Monatsh. Math. 143, No. 3, 229--245 (2004; Zbl 1162.11361)] states that for any real number \(\xi\), and every prime number \(p\), \[ \inf_{q \in \mathbb{Z}, q \ge 1} q \cdot \Vert q\xi \Vert \cdot \vert q \vert_p = 0, \] where \(\Vert \cdot \Vert\) denotes the distance to the nearest integer and \(\vert \cdot \vert_p\) denotes the \(p\)-adic absolute value. A related conjecture states that for every prime number \(p\) and every \(p\)-adic number \(x\), \[ \inf_{a,b \in \mathbb{Z}\setminus\{0\}} \vert ab\vert \cdot \vert ax-b\vert_p = 0. \] Both conjectures remain open, though it is known from the work of \textit{M. Einsiedler} and \textit{D. Kleinbock} [Compos. Math. 143, No. 3, 689--702 (2007; Zbl 1149.11036)] that the set of elements for which either conjecture fails must lie in a countable union of sets of box counting dimension \(0\). In the present paper, the authors are concerned with the second conjecture and the relation between the two. The first two results are metric in nature. Letting \(\log^*(n) = \max\{1, \log n\}\), it is shown that a strong form of the second conjecture holds for almost all \(x \in \mathbb{Q}_p\) with respect to the Haar measure, namely that for almost all \(x \in \mathbb{Q}_p\), \[ \inf_{a,b \in \mathbb{Z}\setminus\{0\}} \vert ab\vert \cdot \log^* \vert ab\vert \cdot \log^* \log^* \vert ab\vert \cdot \left\vert x-\frac{a}{b} \right\vert_p = 0. \] Subsequently, it is shown that this result fails for a set of maximal Hausdorff dimension. Secondly, a necessary and sufficient criterion for the second conjecture to hold for a \(p\)-adic number \(x \in \mathbb{Q}_p\) is derived in terms of the continued fractions of the truncated and suitably normalised Hensel expansions of \(x\). It is then shown that under the original \(p\)-adic Littlewood conjecture, this condition is satisfied, so that the first conjecture implies the second. Towards the end of the…
reviewed by: Simon Kristensen
Mathematics Subject Classification ID: 11J04
Mathematics Subject Classification ID: 11J61
Mathematics Subject Classification ID: 11J83
zbMATH DE Number: 7488516
zbMATH Keywords: Diophantine approximation
zbMATH Keywords: \(p\)-adic numbers
zbMATH Keywords: Haar measure
zbMATH Keywords: multiplicative problems
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/20195785
OpenAlex ID: W3114277456
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DOI: 10.1307/MMJ/20195785
description: scientific article; zbMATH DE number 7488516