ON HOMOGENEOUS AND INHOMOGENEOUS DIOPHANTINE APPROXIMATION OVER THE FIELDS OF FORMAL POWER SERIES

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 a which ensures that, for some positive epsilon, the set lim inf (Q is an element of Fq[z], deg Q ->infinity )parallel to Q parallel to. min(y is an element of Fq[z]) parallel to Q proportional to - theta -y parallel to >= epsilon has full Hausdorff dimension.