Vol. 302, No. 2, 2019

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On homogeneous and inhomogeneous Diophantine approximation over the fields of formal power series

Yann Bugeaud and Zhenliang Zhang

Vol. 302 (2019), No. 2, 453–480
Abstract

We prove over fields of power series the analogues of several Diophantine approximation results obtained over the field of real numbers. In particular we establish the power series analogue of Kronecker’s theorem for matrices, together with a quantitative form of it, which can also be seen as a transference inequality between uniform approximation and inhomogeneous approximation. Special attention is devoted to the one-dimensional case. Namely, we give a necessary and sufficient condition on an irrational power series α which ensures that, for some positive ε, the set

liminf QFq[z], deg QQ minyFq[z]Qα θ y ε

has full Hausdorff dimension.

Keywords
Diophantine approximation, power series field, exponent of homogeneous approximation, exponent of inhomogeneous approximation, Hausdorff dimension
Mathematical Subject Classification 2010
Primary: 11K55
Secondary: 11J04, 28A80
Milestones
Received: 10 January 2019
Revised: 28 March 2019
Accepted: 1 April 2019
Published: 27 November 2019
Authors
Yann Bugeaud
IRMA UMR 7501
Université de Strasbourg, CNRS
Strasbourg
France
Zhenliang Zhang
School of Mathematical Sciences
Henan Institute of Science and Technology
Xinxiang
China