New York Journal of Mathematics
Volume 26 (2020), 362-377


Dzmitry Badziahin and Yann Bugeaud

On simultaneous rational approximation to a real number and its integral powers, II

view    print

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

For a positive integer n and a real number ξ, let λn(ξ) denote the supremum of the real numbers λ for which there are arbitrarily large positive integers q such that ||qξ||, ||qξ2||, ..., ||qξn|| are all less than q. 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 λn(ξ) is equal (or greater than or equal) to a given value.


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


Yann Bugeaud:
Université de Strasbourg
Mathématiques, 7 rue René Descartes
67084 STRASBOURG, France