A Further Digression into Number Theory: Theorems of Roth and Khinchin

Abstract

This chapter is our second digression into number theory. We saw in §1.8 that Jensen’s Formula (or the First Main Theorem) can be viewed as an analogue of the Artin-Whaples product formula in number theory. In the present chapter, we discuss a celebrated number theory theorem known as Roth’s Theorem, and we explain how this is analogous to a weak form of the Second Main Theorem. In fact, it was the formal similarity between Nevanlinna’s Second Main Theorem and Roth’s Theorem that led C. F. Osgood (see [Osg 1981] and [Osg 1985]) to the discovery of an analogy between Nevanlinna theory and Diophantine approximations. As we said previously, such an analogy was later, but independently, explored in greater depth by P. Vojta. In this chapter we will discuss, without proof, some of the key points in Vojta’s monograph [Vojt 1987], and we include Vojta’s so called “dictionary” relating Nevanlinna theory and number theory. This section is intended for the analytically inclined and is only intended to provide the most basic insight into the beautiful analogy between Nevanlinna theory and Diophantine approximation theory. By omitting proofs, we have tried to make this section less demanding on the reader than [Vojt 19871. However, a true appreciation for Vojta’s analogy cannot be obtained without also studying the proofs of the number theoretic analogues in their full generality. Any reader that is seriously interested in the connection between Nevanlinna theory and Diophantine approximation is highly encouraged to carefully read Vojta’s monograph [Vojt 1987].