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On the complexity of a putative counterexample to the $p$-adic Littlewood conjecture

Part of: Dynamical systems with hyperbolic behavior Diophantine approximation, transcendental number theory Ergodic theory

Published online by Cambridge University Press:  19 May 2015

Dmitry Badziahin ,
Yann Bugeaud ,
Manfred Einsiedler  and
Dmitry Kleinbock
Dmitry Badziahin
Affiliation:
University of Durham, Department of Mathematical Sciences, South Rd, Durham DH1 3LE, UK email dzmitry.badziahin@durham.ac.uk
Yann Bugeaud
Affiliation:
Université de Strasbourg, Mathématiques, 7 rue René Descartes, 67084 Strasbourg Cedex, France email bugeaud@math.unistra.fr
Manfred Einsiedler
Affiliation:
ETH Zürich, Departement Mathematik, Rämistrasse 101, CH-8092 Zürich, Switzerland email manfred.einsiedler@math.ethz.ch
Dmitry Kleinbock
Affiliation:
Brandeis University, Department of Mathematics, Waltham, MA 02454, USA email kleinboc@brandeis.edu
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Abstract

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Let $\Vert \cdot \Vert$ denote the distance to the nearest integer and, for a prime number $p$, let $|\cdot |_{p}$ denote the $p$-adic absolute value. Over a decade ago, de Mathan and Teulié [Problèmes diophantiens simultanés, Monatsh. Math. 143 (2004), 229–245] asked whether $\inf _{q\geqslant 1}$$q\cdot \Vert q{\it\alpha}\Vert \cdot |q|_{p}=0$ holds for every badly approximable real number ${\it\alpha}$ and every prime number $p$. Among other results, we establish that, if the complexity of the sequence of partial quotients of a real number ${\it\alpha}$ grows too rapidly or too slowly, then their conjecture is true for the pair $({\it\alpha},p)$ with $p$ an arbitrary prime.

Keywords

Diophantine approximationLittlewood conjecturecomplexitycontinued fractionsmeasure rigidity

MSC classification

Primary: 11J04: Homogeneous approximation to one number
Secondary: 11J61: Approximation in non-Archimedean valuations 11J83: Metric theory 37A35: Entropy and other invariants, isomorphism, classification 37A45: Relations with number theory and harmonic analysis 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 9 , September 2015 , pp. 1647 - 1662
Copyright
© The Authors 2015 

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