Diophantine Approximation with Algebraic Points of Bounded Degree

✍ Scribed by Min Ru; Julie Tzu-Yueh Wang

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 158 KB
• Volume: 81
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1999.2462

✦ Synopsis

In this paper, we extend Schmidt's subspace theorem to the approximation of algebraic numbers by algebraic points of bounded degree. 2000 Academic Press 1. INTRODUCTION This paper deals with the Diophantine approximation with algebraic points of bounded degree. In 1955, Roth proved his celebrated theorem that for any real algebraic number :, and every }>2, the inequality |(x y)&:| >(1Âmax[ |x|, | y| ] } ) holds for all, but finitely many (x y) # Q.

Roth's theorem was extended by E. Wirsing in 1968 to the approximation of the algebraic number : by algebraic numbers of bounded degree. In 1970, W. Schmidt generalized Roth's theorem to the approximation of more than one algebraic number by rational points. In this paper, we extend Schmidt's theorem to the approximation by algebraic points of bounded degree. However, we could not fully obtain Wirsing's version of Schmidt's subspace theorem. This paper thus could be regarded as a first step towards the complete understanding of Diophantine approximation with algebraic points of bounded degree. We note that this paper is motivated by the recent discovered relationship between Diophantine approximation and Nevanlinna theory. See [Ru] for the further discussion.