We turn to the study of a class of generating functions that, somewhat like the folds and ripples of the previous chapter, lead to remarkable continued fractions and rational approximations. They rely on an inhomogeneous continuous function algorithm discussed below.

Inhomogeneous Diophantine approximation

As we have already learnt, the convergents pn/qn of a continued fraction [a0; a1, a2, …] representing the real irrational number α minimise the quantity |qα−p|. The related homogeneous Diophantine problem initiated by Dirichlet's theorem (Theorem 1.36) and discussed in Sections 1.4 and 1.5 was in fact our motivation to develop the theory of continued fractions.

It is not therefore completely unreasonable to believe that a slightly more general minimisation problem for the quantity |qα + β − p|, where β is another (not necessarily irrational) real number already considered in Chebyshev's theorem (Exercise 1.39), gives rise to a natural extension of continued fractions. More important is not so much the algorithmic solution of this inhomogeneous Diophantine approximation problem but its many consequences.

Because the replacement of α and β by their fractional parts does not affect the Diophantine problem, we will assume that they lie between 0 and 1.