ABSTRACT
In this chapter,we assume the background on continued fractions, rational approximations, quadratic irrationals, and related topics covered, for instance, in [68, Chapter 5].
We have already looked at some Diophantine equations in §1.1. In particular, in Definition 1.10 on page 13, and Theorem 1.8 on page 14, we considered the Ramanujan-Nagell equation, the generalization of which we will study later in the text. The relationship between the solution of Diophantine equations and approximation of algebraic numbers by rational numbers is the focus of this section. In particular, we know from [68, Corollary 5.3, p. 215, Exercise 5.10, p.220], for instance, that there are infinitely many rational number p, q such that ∣∣∣∣α− pq
∣∣∣∣ < 1q2 . (4.1) A natural query is: Can the exponent 2 be increased to get a general result that improves upon (4.1)? In a drive to answer this question, the Fields medal was achieved by Roth in 1958 for his 1955 result: If α is an algebraic number, then for a given ε > 0, there exist at most finitely many rational numbers p, q, with
∣∣∣∣α− pq ∣∣∣∣ < 1q2+ε (4.2)
–see [21]. Roth’s work was preceded by results of Thue in 1909 and Siegel in 1921-see [68, Biography 1.12, p. 45] and Biography 4.4 on page 170. Both of the latter two improved upon the following result of Liouville-see Biography 4.3 on page 168.
