Approximation by rational numbers

Abstract

Throughout the present Chapter, we are essentially concerned with the following problem: for which functions Ψ : ℝ ≥1 → : ℝ ≥0 is it true that, for a given real number ξ, or for all real numbers ξ in a given class, the equation |ξ – p/q | q ) has infinitely many solutions in rational numbers p/q ? We begin by stating the results on rational approximation obtained by Dirichlet and Liouville in the middle of the nineteenth century. In Section 1.2, we define the continued fraction algorithm and recall the main properties of continued fractions expansions. These are used in Section 1.3 to give a full proof of a metric theorem of Khintchine. The next two Sections are devoted to the Duffin–Schaeffer Conjecture and to some complementary results on continued fractions. Dirichlet and Liouville Every real number ξ can be expressed in infinitely many ways as the limit of a sequence of rational numbers. Furthermore, for any positive integer b , there exists an integer a with |ξ – a/b | ≤ 1/(2 b ), and one may hope that there are infinitely many integers b for which |ξ – a/b | is in fact much smaller than 1/(2 b ). For instance, this is true when ξ is irrational, as follows from the theory of continued fractions.