Simultaneous Diophantine approximation

The simplest problems of Diophantine approximation relate to the approximation of a single irrational number 0 by rational numbers pjq, and the principal question is how small we can make the error 0 — pjq in relation to q for infinitely many approximations. It is well known that this question can be answered almost completely in terms of the continued fraction expansion of 0. It must be admitted that our knowledge of the relationship between the continued fraction expansion of 0 and other possible representations of 0 is very fragmentary, a striking enough example being the number tf/2. However, the continued fraction theory gives us many general results which are best possible. Thus every 0 admits an infinity of approximations satisfying