Abstract:
In 1949, Mahler [9] started the study of rational approximation to algebraic elements in the field of power series with coefficients in a finite
field,
by adapting a classical theorem of Liouville concerning rational approximation to algebraic real numbers. In his paper, Mahler pointed at the difference
with classical case by introducing an example. This example, having rational approximation reaching the Liouville's bound, proves that we haven't an analogue of
Thue-Siegel-Roth theorem
in positive characteristic. In this paper, we will prove that this example encompasses a large family of power series that are elements of Pisot and
Salem retaining the same property of well approximation by the rationals. Further, we will discuss some Diophantine equations, closely related to these elements.
Key Words: Finite fields, formal power series,
continued fraction.
2010 Mathematics Subject Classification: Primary 11J61; Secondary 11J70.
