Vol. 9, No. 4, 2020
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Effective simultaneous rational approximation to pairs of real quadratic numbers

Yann Bugeaud
Vol. 9 (2020), No. 4, 353–360
DOI: 10.2140/moscow.2020.9.353
Abstract

Let ξ, ζ be quadratic real numbers in distinct quadratic fields. We establish the existence of effectively computable, positive real numbers τ and c such that, for every integer q with q > c, we have

max{∥qξ∥,∥qζ∥} > q−1+τ,

where ∥⋅∥ denotes the distance to the nearest integer.

To the memory of Naum Ilich Feldman (1918–1994)

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Keywords
simultaneous approximation, Pell equation, linear form in logarithms
Mathematical Subject Classification 2010
Primary: 11J13
Secondary: 11D09, 11J86
Milestones
Received: 29 July 2019
Revised: 9 March 2020
Accepted: 23 March 2020
Published: 5 November 2020
Authors
Yann Bugeaud
Institut de Recherche Mathématique Avancée, UMR 7501
Université de Strasbourg et CNRS
Strasbourg
France