Vol. 11, No. 1, 2022

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Exponents of Diophantine approximation in dimension 2 for a general class of numbers

Anthony Poëls

Vol. 11 (2022), No. 1, 37–69
Abstract

We study the Diophantine properties of a new class of transcendental real numbers which contains, among others, Roy’s extremal numbers, Bugeaud–Laurent Sturmian continued fractions, and more generally the class of Sturmian-type numbers. We compute, for each real number ξ of this set, several exponents of Diophantine approximation to the pair (ξ,ξ2), together with ω2(ξ) and ω^2(ξ), the so-called ordinary and uniform exponents of approximation to ξ by algebraic numbers of degree 2. As an application, we get new information on the set of values taken by ω^2 at transcendental numbers, and we give a partial answer to a question of Fischler about his exponent β0.

Keywords
exponents of approximation, parametric geometry of numbers, approximation by algebraic numbers, simultaneous approximation
Mathematical Subject Classification
Primary: 11J13
Secondary: 11H06, 11J82
Milestones
Received: 12 July 2021
Revised: 10 October 2021
Accepted: 25 October 2021
Published: 30 March 2022
Authors
Anthony Poëls
Department of Mathematics
University of Ottawa
Ottawa
Canada