In Chapters 1 to 7, we have exclusively considered approximation of real numbers. However, Mahler [376] and Koksma [333] defined their classifications for complex numbers as well, and Mahler [378] also introduced an analogous classification for the transcendental numbers in the field ℚp, the completion of ℚ with respect to the prime number p. Furthermore, approximation in the field of formal power series has also been investigated, for example, by Sprindžuk [534, 539]. In the present Chapter, we consider each of these settings, and we briefly describe the state of the art for the problems corresponding to those studied in Chapters 1 to 7. Roughly speaking, it is believed (and it often turns out to be true) that Diophantine approximation results in the real case have got their complex and p-adic analogues, the proofs of which are a (more or less) straightforward adaptation of those in the real case. This however does not hold true anymore for Diophantine approximation in fields of power series. For instance, the analogue of Roth's Theorem 2.1 does not exist when the ground field has positive characteristic, see, for example, the surveys by Lasjaunias [351] and by Schmidt [515] for additional information.

Approximation in the field of complex numbers

Let ξ be a complex non-real number and let n be a positive integer.