Diophantine approximation and the weighted products of partial quotients in continued fractions

16 June 2022

Abstract

The fundamental objective of the theory of Diophantine approximation is to seek answer to a simple question “how well irrational numbers can be approximated by rational numbers?”. In this regard the theory of continued fractions provides quick and efficient way for finding good rational approximations to irrational numbers.  

In this talk, first I will discuss the relationship between Diophantine approximation and the theory of continued fractions. I relate the three fundamental theories in metric Diophantine approximation (Dirichlet’s theorem, Khintchine’s theorem and Jarnik’s theorem) to the questions in continued fractions. Then I will describe some metrical properties of the product of consecutive partial quotients raised to different powers in continued fractions.