 

Dzmitry Badziahin and
Yann Bugeaud
On simultaneous rational approximation to a real number and its integral powers, II
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print


Published: 
April 8, 2020. 
Keywords: 
simultaneous approximation, transference theorem. 
Subject: 
11J13. 


Abstract
For a positive integer n and a real number ξ, let λ_{n}(ξ) denote the supremum of the real numbers λ for which there are arbitrarily large positive integers q such that qξ, qξ^{2}, ..., qξ^{n} are all less than q^{λ}. Here,   denotes the distance to the nearest integer. We establish new results on the Hausdorff dimension of the set of real numbers ξ such that λ_{n}(ξ)
is equal (or greater than or equal) to a given value. 

Acknowledgements
The main part of this work has been done
while Yann Bugeaud was visiting the University of Sydney, supported by the Sydney Mathematical Research Institute International Visitor Program. The authors are grateful to the referee for a very careful reading.


Author information
Dzmitry Badziahin:
The University of Sydney
Camperdown 2006, NSW, Australia
dzmitry.badziahin@sydney.edu.au
Yann Bugeaud:
Université de Strasbourg
Mathématiques, 7 rue René Descartes
67084 STRASBOURG, France
bugeaud@math.unistra.fr

