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

$\underset{Q\in {\mathbb{F}}_{q}\left[z\right],\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}degQ\to \infty }{liminf}\parallel Q\parallel \cdot \underset{y\in {\mathbb{F}}_{q}\left[z\right]}{min}\parallel Q\alpha -\theta -y\parallel \ge \epsilon$

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