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

DOI: 10.2140/pjm.2019.302.453

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 $\alpha$ which ensures that, for some positive $\epsilon$, the set

$$\underset{Q\in {\mathbb{F}}_q\left[z\right],degQ\to \infty }{liminf}\parallel Q\parallel \cdot \underset{y\in {\mathbb{F}}_q\left[z\right]}{min}\parallel Q\alpha -\theta -y\parallel \ge \epsilon$$

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